On Exact Cover, Efficient Domination and Efficient Edge...

On Exact Cover, Efficient Domination

and Efficient Edge Domination in

Graphs and Hypergraphs

Andreas Brandstädt,

University of Rostock, Germany

(with C. Hundt, A. Leitert, M. Milanič, R. Mosca, R. Nevries, and D. Rautenbach)

University of Rostock was founded in 1419

Based on results of

- [B., Hundt, Nevries LATIN 2010]

- [B., Mosca ISAAC 2011; Algorithmica]

- [B., Leitert, Rautenbach ISAAC 2012]

- [B., Milanič, Nevries MFCS 2013]

Exact Cover by 3-Sets (X3C)

Problem [SP2] of [Garey, Johnson 1979]:

INSTANCE: A finite set X with | X | 3q and a

collection C of 3-element subsets of X.

QUESTION: Does C contain an exact cover

for X, i.e., a subcollection D C such that

every element of X occurs in exactly one

member of D?

a b c d e f

Exact Cover by 3-Sets (X3C)

Theorem [Karp 1972]

X3C is NP-complete.

(reduction from 3DM)

Remark. Exact Cover by 2-Sets corresponds

to Perfect Matching.

Exact Cover

Let H (V,E) be a hypergraph and w(e): |e|.

Fact.

M is an exact cover in H

M is a maximum weight independent set in

L(H) with w(M) |V |.

Efficient domination

Let G (V,E) be a finite undirected graph.

A vertex v dominates itself and its neighbors,

i.e., v dominates N[v].

[Biggs 1973, Bange, Barkauskas, Slater 1988]:

D is an efficient dominating set (e.d.) in G if

(1) it is dominating in G and

(2) every vertex is dominated exactly once.


Not every graph has an efficient dominating

set!


Efficient dominating sets in G are also called

independent perfect dominating sets.

Let G2 (V, E2) with xy E2 if dG(x,y) 2.

Fact. Let N(G) be the closed neighborhood

hypergraph of G. Then:

G2 L(N(G))


Fact. For D V, the following are equivalent:

(1) D is an e.d. in G.

(2) D dominating in G and independent in G2.

(3) the closed neighborhoods N[v], v D, are

an exact cover for V(G).

Corollary. G has an e.d. N(G) has an exact

cover for V(G).


The ED problem:

INSTANCE: A finite graph G (V, E).

QUESTION: Does G have an e.d.?

Theorem [Bange, Barkauskas, Slater 1988]

ED is NP-complete.


Theorem [Yen, Lee 1996]

ED is NP-complete for bipartite graphs and for

chordal graphs.

Corollary. For every k > 2,

ED is NP-complete for Ck -free graphs.

Efficient edge domination

[Grinstead, Slater, Sherwani, Holmes, 1993]:

M E is an efficient edge dominating set

(e.e.d.) in G if

- M is dominating in L(G), and

- every edge of E is dominated exactly once

in L(G), that is, M is an e.d. in L(G).

Not every graph (not every tree !) has an e.e.d.:

G: e1 e2 e3 e4

e5

e1 e2 e3 e4

e5

L(G):


The EED problem:

INSTANCE: A finite graph G (V, E).

QUESTION: Does G have an e.e.d.?

Theorem [Grinstead, Slater, Sherwani,

Holmes, 1993] EED is NP-complete.

Corollary. ED is NP-complete for line graphs,

and thus for claw-free graphs.


The ED problem for F-free graphs:

Recall: If F contains a cycle or claw then ED

is NP-complete on F-free graphs.

Thus, for investigating the remaining cases,

we can restrict F to linear forests.


Fact. Every e.d. is a maximal independent set.

Theorem [Balas, Yu 1989], [Alekseev 1991],

[Farber, Hujter, Tuza 1993], [Prisner 1995]

kP2–free graphs have at most n2k-2 maximal

independent sets.

Corollary. For every fixed k, ED is

polynomial for kP2–free graphs.


Recall:

Theorem [Yen, Lee 1996]

ED is NP-complete for bipartite graphs and for

chordal graphs.

Proof by simple standard reduction from X3C:

a b c d e f

q r s

a b c d e f


P3 + P3 = 2P3

and P7


From the standard reduction, it follows:

Corollary. ED is NP-complete for 2P3 -free

graphs, and thus also for P7 -free graphs.


From the standard reduction, it follows:

Corollary. ED is NP-complete for 2P3 -free

graphs, and thus also for P7 -free graphs.

Theorem [B., Milanič, Nevries MFCS 2013]

ED is solvable in time O(n m) for (P2 + P4) -

free graphs and for P5 -free graphs.


G is a split graph if V(G) is partitionable into a

clique and an independent set.

Theorem [Földes, Hammer 1977]

G is a split graph G is (2P2 ,C4 ,C5)–free.

Theorem [M.-S. Chang, Liu 1993]

ED in time O(n + m) for split graphs.

thin spider


Lemma [B., Milanič, Nevries MFCS 2013]

A prime 2P2 -free graph has an e.d. it is a

thin spider.


ED is solvable in time O(n + m) for 2P2 -free

graphs.


Recall: D is an e.d. in G D is dominating in

G and independent in G2.

Let w(v): |N [v]|. Then:

(i) D dominating in G |V | w(D).

(ii) D independent set in G2 w(D) |V |.


Recall: w(v): |N [v]|.

Fact [Leitert; Milanič 2012]

D is an e.d. in G D is a maximum weight

independent set in G2 with w(D) |V |.


Corollary. Let C be a graph class. If the

MWIS problem is solvable in polynomial time

for G2 , for all G C, then ED is solvable in

polynomial time on C.

Examples:

dually chordal graphs: squares are chordal.

AT-free graphs: squares are co-comparability.


Theorem [Milanič 2013]

If G is E– and net–free then G2 is claw–free.



If G is P5–free and has an e.d. then G2 is P4–

free.

Corollary. ED is solvable in time O(MM) for

P5–free graphs.


Summarizing:

- For every F with 5 vertices, the complexity

of ED for F–free graphs is known.

- For F with 6 vertices, the only open case is

the P6.

Open. What is the complexity of ED for P6–

free graphs?

ED NP-c.

P4–free P5–free P6–free P7–free

O(n m) LIN ?

ED

EED

NP-c.

MIS ?

P4–free P5–free P6–free P7–free

O(n m)

LIN

LIN

LIN

LIN

?

LIN LIN

?

?


Efficient edge dominating sets are also called

dominating induced matchings (d.i.m.):

Fact. M E is an e.e.d. in G

(1) the pairwise distance of edges in M is at

least 2 (i.e., M is an induced matching), and

(2) for every edge e E – M, there is exactly

one f M intersecting e.


Recall: D is an e.e.d. in G D is dominating

in L(G) and independent in L(G)2

(i.e., D is an e.d. in L(G)).

Let w(e): |N [e]| (neighborhood w.r.t. L(G)).

Fact. M is an e.e.d. in G

M is a maximum weight independent set in

L(G)2 with w(M) | E |.


Corollary. If the MWIS problem is solvable in

polynomial time for squares of line graphs of a

graph class C then EED is solvable in

polynomial time on C.

Example: weakly chordal graphs

[Cameron, Sritharan, Tang 2003]:

G weakly chordal L(G)2 weakly chordal.

Squares of Line Graphs

• G chordal L(G)2 chordal [Cameron 1989]

• G circular-arc L(G)2 circular-arc [Golumbic,

Laskar 1993]

• G co-comparability L(G)2 co-comparability

[Golumbic, Lewenstein 2000]

• G weakly chordal L(G)2 weakly chordal

[Cameron, Sritharan, Tang 2003]

• stronger result for AT-free graphs [J.-M. Chang

2004]


Theorem [Lu, Tang 1998, Lu, Ko, Tang 2002]

EED is NP-complete for bipartite graphs, and

is solvable in linear time for bipartite

permutation graphs, generalized series-parallel

graphs and for chordal graphs.


Theorem [Lu, Tang 1998, Lu, Ko, Tang 2002]

EED is NP-complete for bipartite graphs, and

is solvable in linear time for bipartite

permutation graphs, generalized series-parallel

graphs and for chordal graphs.

Open [Lu, Ko, Tang 2002]

Complexity of EED for weakly chordal graphs

and for permutation graphs.


Theorem [Cardoso, Lozin 2008]

EED is NP-complete for (very special)

bipartite graphs, and is solvable in polynomial

time for claw-free graphs as well as in linear

time for chordal graphs.


Open [Cardoso, Korpelainen, Lozin 2011]

Complexity of EED for

- Pk–free graphs, k > 4

- chordal bipartite graphs

- weakly chordal graphs


Theorem [B., Hundt, Nevries LATIN 2010]

EED is solvable in

- linear time for chordal bipartite graphs,

- polynomial time for hole-free graphs, and

- is NP-complete for planar bipartite graphs

with maximum degree 3.


Theorem [B., Mosca ISAAC 2011;

Algorithmica]

EED in linear time for P7 -free graphs in a

robust way.

ED for hypergraphs

H = (V,E) - a finite hypergraph.

D V is an e.d. in H if D is an e.d. in 2sec(H).

Thm [B., Leitert, Rautenbach ISAAC 2012]

ED is

- NP-complete for –acyclic hypergraphs, and

- polynomial time for hypertrees.

EED for hypergraphs


M E is an e.e.d. in H if for all e E, there is

exactly one f M intersecting e (possibly fe).

Fact. M is an e.e.d. in H M is an e.d. in

L(H).

Corollary. D is an e.d. in H D is an e.e.d.

in H*. (since 2sec(H) ~ L(H*) )

EED for hypergraphs


EED is

- polynomial for –acyclic hypergraphs, and

- NP-complete for hypertrees.

Maximum induced matchings for

hypergraphs


M E is an induced matching in H if M is an

independent node set in L(H)2.


MIM is

- polynomial for –acyclic hypergraphs, and

- NP-complete for hypertrees.

Exact Cover for hypergraphs


Exact Cover is

- NP-complete for –acyclic hypergraphs, and

- polynomial for hypertrees.

ED

EED

chordal dually chordal

NP-c. [ ] lin.

lin. [ ]

MIM NP-c. pol. [ ]

–acyclic hyp. hypertrees

XC

NP-c.

NP-c.

NP-c.

lin. pol.

pol.

pol.

pol.

NP-c.

[B., Leitert, Rautenbach, ISAAC 2012]:

Thank you for your attention!


Corollary. ED is solvable in polynomial time

for dually chordal graphs and thus also for

strongly chordal graphs.


Theorem [Lu, Tang 2002]

ED is NP-complete for chordal bipartite

graphs.

(proof by complicated reduction from 1-in-3

3SAT)


Open [Lu, Tang 2002]

Complexity of ED for convex bipartite graphs

and for strongly chordal graphs.

Recall:

G strongly chordal G dually chordal

G convex bipartite G interval bigraph

G chordal bipartite


Theorem [Bui-Xuan, Telle, Vatshelle, 2011]

If for a graph class, boolean width is at most

O(log n) then the Minimum Weight

Dominating Set problem can be solved in

polynomial time.

Theorem [Keil, 2012] Boolean width of

interval bigraphs is at most 2 log n.


Corollary.

For interval bigraphs, ED can be solved in

polynomial time.

Recall:

G convex bipartite G interval bigraph


Theorem [B., Milanič, Nevries, MFCS 2013]

ED is NP-complete for planar bipartite graphs

with maximum degree 3.


ED in Monadic Second Order Logic:

Fact. G = (V, E) has an e.d.

V V v V ! v V (v N[v])


EED in Monadic Second Order Logic:

Fact. G = (V, E) has an e.e.d.

E E e E ! e E (e e )

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