The Maximum Induced Planar Subgraph...
Transcript of The Maximum Induced Planar Subgraph...
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The Maximum Induced Planar Subgraph problem
Graham Farr
Faculty of ITMonash University
6 July 2007
Joint work with Keith Edwards (Dundee) and Kerri Morgan
![Page 2: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/2.jpg)
The problem
MAXIMUM INDUCED PLANAR SUBGRAPH (MIPS)Input: Graph GOutput: set P ⊆ V (G ) such that
I the induced subgraph 〈P〉 is planar,
I |P| is maximum.
![Page 3: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/3.jpg)
The problem
MAXIMUM INDUCED PLANAR SUBGRAPH (MIPS)Input: Graph GOutput: set P ⊆ V (G ) such that
I the induced subgraph 〈P〉 is planar,
I |P| is maximum.
![Page 4: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/4.jpg)
The problem
MAXIMUM INDUCED PLANAR SUBGRAPH (MIPS)Input: Graph GOutput: set P ⊆ V (G ) such that
I the induced subgraph 〈P〉 is planar,
I |P| is maximum.
![Page 5: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/5.jpg)
The problem
MAXIMUM INDUCED PLANAR SUBGRAPH (MIPS)Input: Graph GOutput: set P ⊆ V (G ) such that
I the induced subgraph 〈P〉 is planar,
I |P| is maximum.
![Page 6: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/6.jpg)
The problem
MAXIMUM INDUCED PLANAR SUBGRAPH (MIPS)Input: Graph GOutput: set P ⊆ V (G ) such that
I the induced subgraph 〈P〉 is planar,
I |P| is maximum.
![Page 7: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/7.jpg)
Complexity of MIPS
MIPS is
I NP-hard to solve exactly(Krishnamoorthy & Deo, 1979; Lewis & Yannakakis, 1980)
I also hard to approximate (Lund & Yannakakis, 1993):∃ε > 0: cannot get performace ratio n−ε unless P = NP.
I approximable with performance ratio Ω(n−1(log n/ log log n)2)(Halldorsson, 2000)
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Bounded degree MIPS
“Real” graphs have low degrees.
Approximation algorithms: max degree ≤ d :
I Halldorsson & Lau, 1997:proportion of vertices included:
1
d(d + 1)/3eI linear timeI subgraphs found have max degree ≤ 2
I Edwards & Farr, GD 2001:proportion of vertices included:
3
d + 1I time O(mn)I subgraphs found are series-parallel
![Page 9: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/9.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithms
I independent set (induced null subgraph)I induced forestI induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 10: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/10.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)
I induced forestI induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 11: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/11.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)I induced forest
I induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 12: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/12.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)I induced forestI induced series-parallel subgraph
I induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 13: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/13.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)I induced forestI induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 14: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/14.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)I induced forestI induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 15: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/15.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)I induced forestI induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 16: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/16.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)I induced forestI induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 17: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/17.jpg)
Today:
I Max degree ≤ d : some vertex addition algorithmsI independent set (induced null subgraph)I induced forestI induced series-parallel subgraphI induced outerplanar subgraph
I Average degree ≤ d : vertex removal algorithm
I Experiments
I Connection with fragmentability
I Future work
![Page 18: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/18.jpg)
Finding an Independent Set (classical heuristic)
Input: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 0 in P, move it to P.Output: P
Stops when every vertex in R has degree ≥ 1 in P.Count E (P,R) from each side:
d |P| ≥ |R|d |P| ≥ n − |P|
(d + 1)|P| ≥ n
|P| ≥ n
d + 1
Proportion:1
d + 1(Turan)
![Page 19: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/19.jpg)
Finding an Independent Set (classical heuristic)
Input: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 0 in P, move it to P.Output: P
Stops when every vertex in R has degree ≥ 1 in P.
Count E (P,R) from each side:
d |P| ≥ |R|d |P| ≥ n − |P|
(d + 1)|P| ≥ n
|P| ≥ n
d + 1
Proportion:1
d + 1(Turan)
![Page 20: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/20.jpg)
Finding an Independent Set (classical heuristic)
Input: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 0 in P, move it to P.Output: P
Stops when every vertex in R has degree ≥ 1 in P.Count E (P,R) from each side:
d |P| ≥ |R|d |P| ≥ n − |P|
(d + 1)|P| ≥ n
|P| ≥ n
d + 1
Proportion:1
d + 1(Turan)
![Page 21: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/21.jpg)
Finding an Independent Set (classical heuristic)
Input: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 0 in P, move it to P.Output: P
Stops when every vertex in R has degree ≥ 1 in P.Count E (P,R) from each side:
d |P| ≥ |R|d |P| ≥ n − |P|
(d + 1)|P| ≥ n
|P| ≥ n
d + 1
Proportion:1
d + 1(Turan)
![Page 22: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/22.jpg)
Finding an Induced ForestInput: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 1 in
the non-null portion P1 of
P,move it to P.Output: P
Stops when every vertex in R has degree ≥ 2 in P1.P0 := isolated vertices in 〈P〉; P1 = P \ P0.Count E (P1,R) from each side:
(d − 1)|P1| ≥ 2|R|(d − 1)|P1| ≥ 2n − 2|P|
(d + 1)|P1|+ 2|P0| ≥ 2n
|P| = |P1|+ |P0| ≥ 2n
d + 1
Proportion:2
d + 1(Alon, Mubayi, Thomas, 2001)
![Page 23: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/23.jpg)
Finding an Induced ForestInput: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 1 in the non-null portion P1 of P,move it to P.Output: P
Stops when every vertex in R has degree ≥ 2 in P1.P0 := isolated vertices in 〈P〉; P1 = P \ P0.Count E (P1,R) from each side:
(d − 1)|P1| ≥ 2|R|(d − 1)|P1| ≥ 2n − 2|P|
(d + 1)|P1|+ 2|P0| ≥ 2n
|P| = |P1|+ |P0| ≥ 2n
d + 1
Proportion:2
d + 1(Alon, Mubayi, Thomas, 2001)
![Page 24: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/24.jpg)
Finding an Induced ForestInput: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 1 in the non-null portion P1 of P,move it to P.Output: P
Stops when every vertex in R has degree ≥ 2 in P1.
P0 := isolated vertices in 〈P〉; P1 = P \ P0.Count E (P1,R) from each side:
(d − 1)|P1| ≥ 2|R|(d − 1)|P1| ≥ 2n − 2|P|
(d + 1)|P1|+ 2|P0| ≥ 2n
|P| = |P1|+ |P0| ≥ 2n
d + 1
Proportion:2
d + 1(Alon, Mubayi, Thomas, 2001)
![Page 25: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/25.jpg)
Finding an Induced ForestInput: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 1 in the non-null portion P1 of P,move it to P.Output: P
Stops when every vertex in R has degree ≥ 2 in P1.P0 := isolated vertices in 〈P〉; P1 = P \ P0.Count E (P1,R) from each side:
(d − 1)|P1| ≥ 2|R|(d − 1)|P1| ≥ 2n − 2|P|
(d + 1)|P1|+ 2|P0| ≥ 2n
|P| = |P1|+ |P0| ≥ 2n
d + 1
Proportion:2
d + 1(Alon, Mubayi, Thomas, 2001)
![Page 26: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/26.jpg)
Finding an Induced ForestInput: G = (V ,E )P := ∅, R := VLoop: if v ∈ R has degree ≤ 1 in the non-null portion P1 of P,move it to P.Output: P
Stops when every vertex in R has degree ≥ 2 in P1.P0 := isolated vertices in 〈P〉; P1 = P \ P0.Count E (P1,R) from each side:
(d − 1)|P1| ≥ 2|R|(d − 1)|P1| ≥ 2n − 2|P|
(d + 1)|P1|+ 2|P0| ≥ 2n
|P| = |P1|+ |P0| ≥ 2n
d + 1
Proportion:2
d + 1(Alon, Mubayi, Thomas, 2001)
![Page 27: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/27.jpg)
Finding an Induced Planar SubgraphAlgorithm from Edwards & Farr, 2002 (outline):Input: G = (V ,E )P := ∅, R := VP0 := forest portion of 〈P〉; P1 = P \ P0.Loop: if v ∈ R has dP1(v) ≤ 2,
can either move it to P (increases |P|)or swap it with an appropriate vertex in P.(This swap may require dP0(v) ≤ 1.)
Output: P
Stops when every vertex in R has
dP1(v) ≥ 3, or dP1(v) = 2 and dP0(v) ≥ 2
Count E (P,R), or E (P1,R), from each side.
Obtain: Proportion:3
d + 1
Finds an induced series-parallel subgraph.
![Page 28: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/28.jpg)
Finding an Induced Planar SubgraphAlgorithm from Edwards & Farr, 2002 (outline):Input: G = (V ,E )P := ∅, R := VP0 := forest portion of 〈P〉; P1 = P \ P0.Loop: if v ∈ R has dP1(v) ≤ 2,
can either move it to P (increases |P|)or swap it with an appropriate vertex in P.(This swap may require dP0(v) ≤ 1.)
Output: P
Stops when every vertex in R has
dP1(v) ≥ 3, or dP1(v) = 2 and dP0(v) ≥ 2
Count E (P,R), or E (P1,R), from each side.
Obtain: Proportion:3
d + 1
Finds an induced series-parallel subgraph.
![Page 29: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/29.jpg)
Finding an Induced Planar SubgraphAlgorithm from Edwards & Farr, 2002 (outline):Input: G = (V ,E )P := ∅, R := VP0 := forest portion of 〈P〉; P1 = P \ P0.Loop: if v ∈ R has dP1(v) ≤ 2,
can either move it to P (increases |P|)or swap it with an appropriate vertex in P.(This swap may require dP0(v) ≤ 1.)
Output: P
Stops when every vertex in R has
dP1(v) ≥ 3, or dP1(v) = 2 and dP0(v) ≥ 2
Count E (P,R), or E (P1,R), from each side.
Obtain: Proportion:3
d + 1
Finds an induced series-parallel subgraph.
![Page 30: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/30.jpg)
Finding an Induced Planar SubgraphAlgorithm from Edwards & Farr, 2002 (outline):Input: G = (V ,E )P := ∅, R := VP0 := forest portion of 〈P〉; P1 = P \ P0.Loop: if v ∈ R has dP1(v) ≤ 2,
can either move it to P (increases |P|)or swap it with an appropriate vertex in P.(This swap may require dP0(v) ≤ 1.)
Output: P
Stops when every vertex in R has
dP1(v) ≥ 3, or dP1(v) = 2 and dP0(v) ≥ 2
Count E (P,R), or E (P1,R), from each side.
Obtain: Proportion:3
d + 1
Finds an induced series-parallel subgraph.
![Page 31: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/31.jpg)
Finding an Induced Outerplanar Subgraph
Algorithm from Morgan & Farr, 2007 (outline):Input: G = (V ,E )P := ∅, R := VP1 := union of components of size ≥ 3 of 〈P〉.Firstly: P := maximal induced forest of G ,
then make some easy additions to P.Loop: if v ∈ R has degree ≤ 2 in P1,
can either move it to P (increases |P|)or swap it with an appropriate vertex in P.
Output: P
When stopped, every vertex in R has degree ≥ 3 in P1.Count E (P1,R) from each side.
Obtain: Proportion:3
d + 5/3
![Page 32: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/32.jpg)
Finding an Induced Outerplanar Subgraph
Algorithm from Morgan & Farr, 2007 (outline):Input: G = (V ,E )P := ∅, R := VP1 := union of components of size ≥ 3 of 〈P〉.Firstly: P := maximal induced forest of G ,
then make some easy additions to P.Loop: if v ∈ R has degree ≤ 2 in P1,
can either move it to P (increases |P|)or swap it with an appropriate vertex in P.
Output: P
When stopped, every vertex in R has degree ≥ 3 in P1.
Count E (P1,R) from each side.
Obtain: Proportion:3
d + 5/3
![Page 33: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/33.jpg)
Finding an Induced Outerplanar Subgraph
Algorithm from Morgan & Farr, 2007 (outline):Input: G = (V ,E )P := ∅, R := VP1 := union of components of size ≥ 3 of 〈P〉.Firstly: P := maximal induced forest of G ,
then make some easy additions to P.Loop: if v ∈ R has degree ≤ 2 in P1,
can either move it to P (increases |P|)or swap it with an appropriate vertex in P.
Output: P
When stopped, every vertex in R has degree ≥ 3 in P1.Count E (P1,R) from each side.
Obtain: Proportion:3
d + 5/3
![Page 34: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/34.jpg)
Proportion of vertices removedMax degree ≤ d :
proportion ≤ d − 2
d + 1
− 3(d − bdc)(dde − d)
(d + 1)(bdc+ 1)(dde+ 1)
E & F 2001,2002
2003,2007
d1 2 3 4 5 6 7 8 9
0
1
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Proportion of vertices removedMax degree ≤ d :
proportion ≤ d − 2
d + 1
− 3(d − bdc)(dde − d)
(d + 1)(bdc+ 1)(dde+ 1)
E & F 2001,2002
2003,2007
d1 2 3 4 5 6 7 8 9
0
1
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Proportion of vertices removedMax degree ≤ d :
proportion ≤ d − 2
d + 1
− 3(d − bdc)(dde − d)
(d + 1)(bdc+ 1)(dde+ 1)
E & F 2001,2002
2003,2007
d1 2 3 4 5 6 7 8 9
0
1
![Page 37: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/37.jpg)
Proportion of vertices removedMax degree ≤ d :
proportion ≤ d − 2
d + 1
− 3(d − bdc)(dde − d)
(d + 1)(bdc+ 1)(dde+ 1)
E & F 2001,2002
2003,2007
d1 2 3 4 5 6 7 8 9
0
1
![Page 38: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/38.jpg)
Proportion of vertices removedAve degree ≤ d :
proportion ≤ d − 2
d + 1− 3(d − bdc)(dde − d)
(d + 1)(bdc+ 1)(dde+ 1)
E & F 2001,2002 2003,2007
d1 2 3 4 5 6 7 8 9
0
1
![Page 39: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/39.jpg)
Series-parallel reductions
1. isolated vertex:
delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 40: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/40.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 41: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/41.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 42: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/42.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf:
delete
3. degree 2:
delete
insert(if absent)
![Page 43: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/43.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 44: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/44.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 45: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/45.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 46: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/46.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 47: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/47.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 48: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/48.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 49: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/49.jpg)
Series-parallel reductions
1. isolated vertex: delete
2. leaf: delete
3. degree 2:
delete
insert(if absent)
![Page 50: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/50.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 51: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/51.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 52: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/52.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 53: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/53.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 54: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/54.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 55: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/55.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 56: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/56.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 57: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/57.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NT
Qld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 58: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/58.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NT
Qld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 59: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/59.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 60: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/60.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 61: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/61.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 62: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/62.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 63: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/63.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 64: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/64.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 65: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/65.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 66: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/66.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 67: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/67.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic
. . . so r(G ) isempty
![Page 68: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/68.jpg)
Gdo series-parallel reductions
for as long as possibler(G )
reducedgraph
Example:
G : WA
NT
SA
Qld
NSW
Vic
ACT
Tas
ACT
WA
NTQld
SA
NSW
Tas
Vic . . . so r(G ) isempty
![Page 69: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/69.jpg)
FactG is series-parallel ⇔ r(G ) is empty.
Theorem (Duffin, 1965)
G is series-parallel ⇔ G contains no subdivision of K4.
![Page 70: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/70.jpg)
FactG is series-parallel ⇔ r(G ) is empty.
Theorem (Duffin, 1965)
G is series-parallel ⇔ G contains no subdivision of K4.
![Page 71: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/71.jpg)
FactG is series-parallel ⇔ r(G ) is empty.
Theorem (Duffin, 1965)
G is series-parallel ⇔ G contains no subdivision of K4.
![Page 72: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/72.jpg)
Algorithm 1.
1. Input: Graph G .2. P := V (G ) // vertices to be kept
R := ∅ // vertices to be removed
ρ :=
∑v∈V (r(G))
dr(G)(v)− 2
dr(G)(v) + 1
3. while ( |R| < ρ and r(〈P〉) is nonempty )
w := vertex in P with maximum degree in r(〈P〉)P := P \ wR := R ∪ w
4. Output: 〈P〉.
![Page 73: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/73.jpg)
Theorem (E & F, 2003, 2007)
If G has min degree ≥ 3, then Algorithm 1 finds a series-parallelsubgraph of G, and the number |R(G )| of vertices removedsatisfies
|R(G )| ≤∑
v∈V (G)
d(v)− 2
d(v) + 1.
Proof. Induction on n.
Inductive basis: empty graph(min degree ≥ 3: no vertices of degree 0,1,2).
Now let G be any graph with min degree ≥ 3 . . .
![Page 74: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/74.jpg)
Theorem (E & F, 2003, 2007)
If G has min degree ≥ 3, then Algorithm 1 finds a series-parallelsubgraph of G, and the number |R(G )| of vertices removedsatisfies
|R(G )| ≤∑
v∈V (G)
d(v)− 2
d(v) + 1.
Proof. Induction on n.
Inductive basis: empty graph(min degree ≥ 3: no vertices of degree 0,1,2).
Now let G be any graph with min degree ≥ 3 . . .
![Page 75: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/75.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(
r(
G − w
)
)|
induction: ≤∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 76: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/76.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(
r(
G − w
)
)|
induction: ≤∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 77: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/77.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ +
|R(
r(
G − w
)
)|
induction: ≤∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 78: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/78.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(
r(
G − w
)
)|
induction: ≤∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 79: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/79.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 80: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/80.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction:
≤∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 81: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/81.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 82: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/82.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 83: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/83.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 84: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/84.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 85: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/85.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 86: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/86.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 87: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/87.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 88: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/88.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 89: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/89.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 90: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/90.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 91: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/91.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 92: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/92.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 93: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/93.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 94: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/94.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 95: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/95.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 96: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/96.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 97: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/97.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 98: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/98.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑
v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 99: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/99.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑
v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 100: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/100.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 101: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/101.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 102: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/102.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 103: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/103.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑
v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 104: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/104.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑
v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 105: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/105.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤∑
v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 106: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/106.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 107: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/107.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 108: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/108.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 109: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/109.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (r(G−w))
dr(G−w)(v)− 2
dr(G−w)(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 110: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/110.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)
− 1
− 2
d
G−w
(v)
− 1
+ 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 111: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/111.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
dG−w (v)
− 1
− 2
dG−w (v)
− 1
+ 1+
∑v∈V (G−w−N(w))
dG−w (v)− 2
dG−w (v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 112: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/112.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
dG−w (v)
− 1
− 2
dG−w (v)
− 1
+ 1+
∑v∈V (G−w−N(w))
dG−w (v)− 2
dG−w (v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 113: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/113.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 114: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/114.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 115: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/115.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 116: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/116.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 117: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/117.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 118: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/118.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 119: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/119.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 120: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/120.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 121: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/121.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 122: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/122.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 123: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/123.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 124: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/124.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 125: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/125.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 126: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/126.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)
1
+∑
v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 127: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/127.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 128: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/128.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤ + |R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 129: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/129.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 130: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/130.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 131: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/131.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 132: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/132.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 133: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/133.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 134: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/134.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 135: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/135.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 136: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/136.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 137: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/137.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 138: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/138.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( v )(d( v ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 139: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/139.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−∑
v∈N(w)
d(w)
3
d( w )(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 140: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/140.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w) 3
d( w )(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 141: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/141.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 142: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/142.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 143: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/143.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 144: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/144.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 145: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/145.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 146: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/146.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 147: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/147.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 148: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/148.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 149: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/149.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 150: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/150.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 151: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/151.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 152: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/152.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 153: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/153.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 154: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/154.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 155: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/155.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 156: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/156.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 157: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/157.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 158: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/158.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 159: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/159.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 160: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/160.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1
d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 161: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/161.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 162: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/162.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 163: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/163.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 164: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/164.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 165: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/165.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 166: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/166.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 167: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/167.jpg)
N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
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N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
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N(w) rest of G
w
max deg
|R(G )| ≤
+
|R(r(G − w))|
induction: ≤
∑v∈V (
r(
G−w
)
)
d
r(
G−w
)
(v)− 2
d
r(
G−w
)
(v) + 1
∑v∈N(w)
d
G−w
(v)− 1− 2
d
G−w
(v)− 1 + 1+
∑v∈V (G−w−N(w))
d
G−w
(v)− 2
d
G−w
(v) + 1
∑v∈N(w)
d(v)− 3
d(v)+
∑v∈V (G−w−N(w))
d(v)− 2
d(v) + 1
∑v∈N(w)
(d(v)− 3
d(v)− d(v)− 2
d(v) + 1
)+
∑v∈V (G−w)
d(v)− 2
d(v) + 1
−
∑v∈N(w)
d(w)
3
d( w )
(d( w ) + 1)1 +
∑v∈V (G−w)
d(v)− 2
d(v) + 1d(w)− 2
d(w) + 1
∑v∈V (G)
d(v)− 2
d(v) + 1
![Page 170: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/170.jpg)
Can use this result to analyse behaviour of Algorithm 1 for graphsof given average degree:
Theorem (E & F, 2003, 2007)
If G is has average degree d ≥ 4, or is connected and has averagedegree d ≥ 2, then Algorithm 1 finds and induced planar subgraphof G of at least(
3
d + 1+
3(d − bdc)(dde − d)
(d + 1)(bdc+ 1)(dde+ 1)
)n
vertices.
Time complexity = O(nm).
![Page 171: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/171.jpg)
Experiments
I Algorithms: Independent Set (IS), Induced Forest (T),Halldorsson-Lau (HL), Vertex Addition (VA), Outerplanar(OP2), Vertex Removal (VR), . . .
I n = 20, . . . , 10000
I d = 3, 4, . . . , 9
I random graphs:d-regular (Steger-Wormald),expected average degree d (classical)
I number of graphs of each type:50 (for n ≤ 1000), 20 (for n ≥ 1000)
Further information:
I Morgan & Farr, JGAA, to appear (2007)
I http://www.csse.monash.edu.au/~kmorgan/MIPS.html
![Page 172: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/172.jpg)
Performance versus degree: random d -regular graphs
0.2
0.3
0.4
0.5
0.6
0.7
0.8
3 4 5 6 7 8 9
Prop
ortio
n
Degree
VSR+EPSVSR
VROP1+EPS
OP1PT+EPS
OP2PT
TVAHLIS
![Page 173: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/173.jpg)
Performance versus degree: expected ave. degree d
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
3 4 5 6 7 8 9
Prop
ortio
n
Expected Average Degree
VSR+EPSVSR
VROP1+EPS
OP1PT+EPS
OP2PT
TVAIS
HL
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Performance versus n: random d -regular graphs
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Pro
po
rtio
n
n
VSR+EPSVSR
VROP1+EPS
OP1PT+EPS
OP2PT
TVAHLIS
![Page 175: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/175.jpg)
MIPS and fragmentability
MIPS is useful for breaking graphs into small pieces.Given G , with max/ave degree ≤ d :
1. remove vertices from G to leave induced planar subgraph 〈P〉;2. remove o(n) vertices from 〈P〉 to leave bounded size pieces
(e.g., apply Planar Separator Theorem (Lipton & Tarjan)recursively).
Converely, bounds on fragmentability can give upper bound on sizeof MIPS.
E.g., for d = 3, cannot do better than3
d + 1=
3
4.
For more info on fragmentability:Edwards & Farr (2001, 2007),Haxell, Pikhurko & Thomason (preprint)
![Page 176: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/176.jpg)
MIPS and fragmentability
MIPS is useful for breaking graphs into small pieces.Given G , with max/ave degree ≤ d :
1. remove vertices from G to leave induced planar subgraph 〈P〉;2. remove o(n) vertices from 〈P〉 to leave bounded size pieces
(e.g., apply Planar Separator Theorem (Lipton & Tarjan)recursively).
Converely, bounds on fragmentability can give upper bound on sizeof MIPS.
E.g., for d = 3, cannot do better than3
d + 1=
3
4.
For more info on fragmentability:Edwards & Farr (2001, 2007),Haxell, Pikhurko & Thomason (preprint)
![Page 177: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/177.jpg)
Future work
I Improve lower bound on proportion of vertices in MIPS.
Our best:3
d + 1.
Ceiling:4
d + 1. Consider K4 ≤ Kd+1.
How close to ceiling can we get? Is there a lower ceiling?
I MIPS v MPS (ordinary Maximum Planar Subgraph):in each, most good approximation algorithms give tree-likegraphs. Is this coincidence?Does either of these problems help you solve the other?
I Explain experimental results mathematically.
I Experimental comparison with maximal induced planarsubgraph.
![Page 178: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/178.jpg)
Future work
I Improve lower bound on proportion of vertices in MIPS.
Our best:3
d + 1.
Ceiling:4
d + 1. Consider K4 ≤ Kd+1.
How close to ceiling can we get? Is there a lower ceiling?
I MIPS v MPS (ordinary Maximum Planar Subgraph):in each, most good approximation algorithms give tree-likegraphs. Is this coincidence?Does either of these problems help you solve the other?
I Explain experimental results mathematically.
I Experimental comparison with maximal induced planarsubgraph.
![Page 179: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/179.jpg)
Future work
I Improve lower bound on proportion of vertices in MIPS.
Our best:3
d + 1.
Ceiling:4
d + 1. Consider K4 ≤ Kd+1.
How close to ceiling can we get? Is there a lower ceiling?
I MIPS v MPS (ordinary Maximum Planar Subgraph):in each, most good approximation algorithms give tree-likegraphs. Is this coincidence?Does either of these problems help you solve the other?
I Explain experimental results mathematically.
I Experimental comparison with maximal induced planarsubgraph.
![Page 180: The Maximum Induced Planar Subgraph problemusers.monash.edu/~gfarr/research/slides/Farr-mips-talk.pdfThe Maximum Induced Planar Subgraph problem Graham Farr Faculty of IT Monash University](https://reader034.fdocuments.us/reader034/viewer/2022042403/5f14aa17d3376137ed4e46f6/html5/thumbnails/180.jpg)
Future work
I Improve lower bound on proportion of vertices in MIPS.
Our best:3
d + 1.
Ceiling:4
d + 1. Consider K4 ≤ Kd+1.
How close to ceiling can we get? Is there a lower ceiling?
I MIPS v MPS (ordinary Maximum Planar Subgraph):in each, most good approximation algorithms give tree-likegraphs. Is this coincidence?Does either of these problems help you solve the other?
I Explain experimental results mathematically.
I Experimental comparison with maximal induced planarsubgraph.