IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp...
Transcript of IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp...
![Page 1: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/1.jpg)
Recognizing and Drawing
IC-planar Graphs
Philipp KindermannUniversitat Wurzburg /
FernUniversitat in Hagen
Joint work withFranz J. Brandenburg, Walter Didimo, William S. Evans,
Giuseppe Liotta & Fabrizio Montecchiani
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1-planar Graphs
Planar graphs: Can be drawn without crossings.
![Page 3: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/3.jpg)
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
![Page 4: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/4.jpg)
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
![Page 5: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/5.jpg)
1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges
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1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges
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1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges• Recognition: NP-hard [Grigoriev & Bodlander ALG’07]
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1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges• Recognition: NP-hard [Grigoriev & Bodlander ALG’07]
- for planar graphs + 1 edge [Korzhik & Mohar JGT’13]
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1-planar Graphs
1-planar graphs: Each edge is crossed at most once.
Planar graphs: Can be drawn without crossings.
• ≤ 4n− 8 edges• straight-line: ≤ 4n− 9 edges• Recognition: NP-hard [Grigoriev & Bodlander ALG’07]
- for planar graphs + 1 edge [Korzhik & Mohar JGT’13]
- with given rotation system [Auer et al. JGAA’15]
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RAC Graphs
RAC graphs: Can be drawn straight-linewith only right-angle crossings.
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RAC Graphs
RAC graphs: Can be drawn straight-linewith only right-angle crossings.
![Page 12: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/12.jpg)
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
Can be drawn straight-linewith only right-angle crossings.
![Page 13: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/13.jpg)
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
... even for planar graphs [van Krefeld GD’11]
Can be drawn straight-linewith only right-angle crossings.
![Page 14: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/14.jpg)
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
... even for planar graphs [van Krefeld GD’11]
• ≤ 4n− 10 edges [Didimo et al. WADS’09]
Can be drawn straight-linewith only right-angle crossings.
![Page 15: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/15.jpg)
RAC Graphs
• Increases readability
RAC graphs:
[Huang et al. PacificVis’08]
... even for planar graphs [van Krefeld GD’11]
• ≤ 4n− 10 edges [Didimo et al. WADS’09]
Can be drawn straight-linewith only right-angle crossings.
• Recognition: NP-hard [Argyriou et al. JGAA’12]
![Page 16: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/16.jpg)
1-planar RAC graphs
1-planar
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1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
RAC
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1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
? RAC
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1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
RAC
outer-1-planar
• outer-1-planar ⊂ RAC [Dehkordi & Eades IJCGA’12]
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1-planar RAC graphs
1-planar
• 1-planar 6= RAC [Eades & Liotta DMA’13]
RAC
outer-1-planar
• outer-1-planar ⊂ RAC [Dehkordi & Eades IJCGA’12]
perfect RAC
• perfect RAC ⊂ 1-planar [Eades & Liotta DMA’13]
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IC-planar Graphs
IC-planar graphs: Each edge is crossed at most once
independent
crossings
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IC-planar Graphs
IC-planar graphs: Each edge is crossed at most onceand each vertex is incident toat most one crossing edge.indepe
ndentcrossin
gs
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IC-planar Graphs
IC-planar graphs: Each edge is crossed at most onceand each vertex is incident toat most one crossing edge.indepe
ndentcrossin
gs
![Page 24: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/24.jpg)
IC-planar Graphs
• ≤ 13n/4− 6 edges
IC-planar graphs: Each edge is crossed at most onceand each vertex is incident toat most one crossing edge.indepe
ndentcrossin
gs
![Page 25: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/25.jpg)
Recognition
Reduction from 1-planarity testing.
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Recognition
Reduction from 1-planarity testing.
uv
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Recognition
Reduction from 1-planarity testing.
uv
![Page 28: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/28.jpg)
Recognition
Reduction from 1-planarity testing.
uv
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Recognition
Reduction from 1-planarity testing.
uv
u
![Page 30: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/30.jpg)
Recognition
Reduction from 1-planarity testing.
uv
u
![Page 31: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/31.jpg)
Recognition
Reduction from 1-planarity testing.
uv
u
![Page 32: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/32.jpg)
Recognition
Testing IC-planarity is NP-hardTheorem.
Reduction from 1-planarity testing.
uv
u
![Page 33: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/33.jpg)
Recognition
Testing IC-planarity is NP-hardTheorem.
Reduction from 1-planarity testing.
uv
Reduction from planar-3SAT
Recognition
![Page 34: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/34.jpg)
Recognition
Testing IC-planarity is NP-hardTheorem.
Reduction from 1-planarity testing.
uv
Reduction from planar-3SAT
Recognition
![Page 35: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/35.jpg)
Recognition
Testing IC-planarity is NP-hardTheorem.
even if the rotation system is given.
Reduction from 1-planarity testing.
uv
Reduction from planar-3SAT
Recognition
![Page 36: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/36.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
![Page 37: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/37.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
![Page 38: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/38.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!Compute extended dual T ∗ of T .
![Page 39: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/39.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
![Page 40: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/40.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
![Page 41: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/41.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
![Page 42: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/42.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
T ∗ :
![Page 43: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/43.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
![Page 44: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/44.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 45: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/45.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 46: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/46.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 47: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/47.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 48: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/48.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 49: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/49.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 50: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/50.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 51: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/51.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
![Page 52: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/52.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
Task: Find a valid routing for each matching edge!
T :
Compute extended dual T ∗ of T .
u
v
T ∗ :
(u, v) ∈ EM
u
v
Routing in T = path of length 3 in T ∗
![Page 53: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/53.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
![Page 54: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/54.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)
![Page 55: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/55.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
![Page 56: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/56.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
![Page 57: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/57.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
X
![Page 58: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/58.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
X
X
![Page 59: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/59.jpg)
Triangulation + Matching
Given a triconnected plane graph T = (V,ET )and a matching M = (V,EM ),is G = (V,ET ∪ EM ) IC-planar?
u
v
luv ruv
Interior I(u, v)The boundaries of twointeriors may not intersect.
X
X
X
×
![Page 60: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/60.jpg)
Triangulation + Matching
![Page 61: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/61.jpg)
Triangulation + Matching
u
v
![Page 62: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/62.jpg)
Triangulation + Matching
u
v
![Page 63: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/63.jpg)
Triangulation + Matching
u
v
![Page 64: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/64.jpg)
Triangulation + Matching
a
b
u
v
![Page 65: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/65.jpg)
Triangulation + Matching
a
b
u
v
![Page 66: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/66.jpg)
Triangulation + Matching
a
b
u
v
![Page 67: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/67.jpg)
Triangulation + Matching
a
b
c d
u
v
![Page 68: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/68.jpg)
Triangulation + Matching
a
b
c d
u
v
![Page 69: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/69.jpg)
Triangulation + Matching
a
b
c d
u
v
![Page 70: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/70.jpg)
Triangulation + Matching
a
b
c d
u
v
Hierarchical structure: Tree H = (VH , EH)
H:
![Page 71: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/71.jpg)
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
Hierarchical structure: Tree H = (VH , EH)VH = {Iuv | (u, v) ∈M}
H:
![Page 72: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/72.jpg)
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
Hierarchical structure: Tree H = (VH , EH)VH = {Iuv | (u, v) ∈M} ∪ {G}
H:
![Page 73: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/73.jpg)
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
Hierarchical structure: Tree H = (VH , EH)
(Iuv, Iab) ∈ EH ⇔ Iuv ⊂ IabVH = {Iuv | (u, v) ∈M} ∪ {G}
H:
![Page 74: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/74.jpg)
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
Hierarchical structure: Tree H = (VH , EH)
(Iuv, Iab) ∈ EH ⇔ Iuv ⊂ Iaboutdeg(Iuv) = 0⇒ (Iuv, G) ∈ EH
VH = {Iuv | (u, v) ∈M} ∪ {G}
H:
![Page 75: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/75.jpg)
Triangulation + Matching
a
b
c d
u
vIcdIab
Iuv
G
![Page 76: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/76.jpg)
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
![Page 77: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/77.jpg)
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
![Page 78: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/78.jpg)
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
![Page 79: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/79.jpg)
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
![Page 80: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/80.jpg)
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
![Page 81: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/81.jpg)
Triangulation + Matching
a
b
c dIcdIab
Iuv
G
![Page 82: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/82.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
c dIcdIab
Iuv
G
![Page 83: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/83.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c dIcdIab
Iuv
G
![Page 84: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/84.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
![Page 85: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/85.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
![Page 86: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/86.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
![Page 87: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/87.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
![Page 88: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/88.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
![Page 89: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/89.jpg)
Triangulation + Matching
• Always pick “middle” routing
a
b
• Solve rest with 2SAT
c d
u
vIcdIab
Iuv
G
• Recursively check which routings are valid
Theorem.IC-planarity can be tested efficiently if the input graph is atriangulated planar graph and a matching
![Page 90: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/90.jpg)
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
![Page 91: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/91.jpg)
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Using a special 1-planar drawing...
[Alam et al. GD’13]
![Page 92: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/92.jpg)
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
![Page 93: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/93.jpg)
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
![Page 94: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/94.jpg)
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Straight-line RAC drawings of IC-planar graphs may requireexponential area.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
![Page 95: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/95.jpg)
Straight-Line Drawings
IC-plane graphs can be drawn straight-lineon the O(n)×O(n) grid in O(n) time.
Theorem.
Straight-line RAC drawings of IC-planar graphs may requireexponential area.
Theorem.
Using a special 1-planar drawing...
RAC?
[Alam et al. GD’13]
![Page 96: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/96.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
![Page 97: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/97.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
[de Fraysseix, Pach & Pollack Comb’90]
![Page 98: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/98.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 99: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/99.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 100: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/100.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 101: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/101.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 102: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/102.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 103: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/103.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 104: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/104.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 105: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/105.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 106: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/106.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Augment to 3-connected planar graph
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
![Page 107: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/107.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
![Page 108: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/108.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
![Page 109: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/109.jpg)
Straight-Line RAC Drawings
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
![Page 110: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/110.jpg)
Straight-Line RAC Drawings
• Adjust step in which d is placed
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
![Page 111: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/111.jpg)
Straight-Line RAC Drawings
• Adjust step in which d is placed
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
Highest number incanonical order
![Page 112: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/112.jpg)
Straight-Line RAC Drawings
• Adjust step in which d is placed
• Remove one edge per crossing
Adjust Shift-Algorithm for planar graphs
• Contour only has slopes ±1
• Insert vertices in canonical order
[de Fraysseix, Pach & Pollack Comb’90]• Augment to planar-maximal IC-planar graph
• Each crossing → Kite K = (a, b, c, d)
d
cb
a
c
a
Al(b) b
![Page 113: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/113.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
![Page 114: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/114.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
![Page 115: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/115.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
![Page 116: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/116.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
Al(b) b
![Page 117: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/117.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
c
a
![Page 118: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/118.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
![Page 119: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/119.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
![Page 120: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/120.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
d
![Page 121: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/121.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b c
a
d
![Page 122: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/122.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) c
a
d
b
![Page 123: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/123.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) b
a
c
![Page 124: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/124.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
![Page 125: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/125.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
![Page 126: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/126.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
![Page 127: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/127.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
![Page 128: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/128.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
Al(b) bu
d
a
c
![Page 129: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/129.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
bu
d
a
c
![Page 130: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/130.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
bu
d
a
c
![Page 131: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/131.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
bu
c
a
![Page 132: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/132.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
b
![Page 133: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/133.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
b
![Page 134: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/134.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
Al(b) b
![Page 135: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/135.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
Al(b) b
![Page 136: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/136.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
u
c
a
Al(b) b
![Page 137: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/137.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) ba
u
![Page 138: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/138.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) b
d
a
u
![Page 139: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/139.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) b
d
a
u
![Page 140: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/140.jpg)
Straight-Line RAC Drawings
Adjust Shift-Algorithm for planar graphs[de Fraysseix, Pach & Pollack Comb’90]
d
cb
a
cAl(b) b
d
a
uIC-planar graphs can be drawn straight-line RAC inexponential area in O(n3) time.
Theorem.
![Page 141: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/141.jpg)
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
![Page 142: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/142.jpg)
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
![Page 143: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/143.jpg)
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
?
![Page 144: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/144.jpg)
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
?
Draw in polynomial area with good crossing resolution?
![Page 145: IC-planar Graphs Recognizing and Drawing...Recognizing and Drawing IC-planar Graphs Philipp Kindermann Universit at W urzburg / FernUniversit at in Hagen Joint work with Franz J. Brandenburg,](https://reader033.fdocuments.us/reader033/viewer/2022060704/6070a1388ebc410767568477/html5/thumbnails/145.jpg)
Conclusion
1-planar RAC
outer-1-planar
perfect RAC
IC-planar
?
Draw in polynomial area with good crossing resolution?
What about maximal IC-planar graphs?