Efficient Domination for Some Subclasses of $P_6$-Free Graphs in Polynomial Time

Brandstädt, Andreas; Eschen, Elaine M.; Friese, Erik

doi:10.48550/arxiv.1503.00091

Cited by 3 publications

(8 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This generalizes WED for S 1,2,2 -free chordal graphs as well as for S 1,1,3 -free chordal graphs (S 1,2,2 and S 1,1,3 are induced subgraphs of extended gem-see Figure 3 and recall Theorem 2) and for P 6 -free chordal graphs (recall [5,6]).…”

supporting

confidence: 61%

“…Motivated by the G 2 approach in [5,6], and the result of Milanič [30] showing that for (S 1,2,2 ,net)free graphs G, its square G 2 is claw-free, we show in this section that G 2 is chordal for H-free chordal graphs with e.d.s. when H is a net or an extended gem (see Figure 3 -extended gem generalizes S 1,2,2 and some other subgraphs), and thus, WED is solvable in polynomial time for these two graph classes.…”

Section: Wed Is Np-complete For Chordal Hereditary Satgraphsmentioning

confidence: 91%

“…While the complexity of ED for 2P 3 -free chordal graphs is NP-complete (as mentioned above), it was shown in [5] that WED is solvable in polynomial time for P 6 -free chordal graphs, since the square of every P 6 -free chordal graph G with e.d.s. is also chordal.…”

Section: Weighted Efficient Domination (Wed)mentioning

confidence: 99%

See 2 more Smart Citations

On Efficient Domination for Some Classes of H-Free Chordal Graphs

Brandstädt

Mosca

2017

Electronic Notes in Discrete Mathematics

Self Cite

View full text Add to dashboard Cite

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted graph classes such as for 2P 3 -free chordal graphs while it is solvable in polynomial time for P 6 -free chordal graphs (and even for P 6 -free graphs). A standard reduction from the NP-complete Exact Cover problem shows that ED is NP-complete for a very special subclass of chordal graphs generalizing split graphs. The reduction implies that ED is NP-complete e.g. for double-gem-free chordal graphs while it is solvable in linear time for gem-free chordal graphs (by various reasons such as bounded clique-width, distance-hereditary graphs, chordal square etc.), and ED is NP-complete for butterfly-free chordal graphs while it is solvable in linear time for 2P 2 -free graphs.We show that (weighted) ED can be solved in polynomial time for H-free chordal graphs when H is net, extended gem, or S 1,2,3 .

show abstract

supporting

confidence: 61%

Section: Wed Is Np-complete For Chordal Hereditary Satgraphsmentioning

confidence: 91%

See 1 more Smart Citation

On Efficient Domination for Some Classes of H-Free Chordal Graphs

Brandstädt

Mosca

2017

Electronic Notes in Discrete Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Many papers have studied the complexity of ED on special graph classes -see e.g. [1,2,4,8] for references. In particular, a standard reduction from the Exact Cover problem shows that ED remains NP-complete for 2P 3 -free (and thus, for P 7 -free) chordal graphs.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, a standard reduction from the Exact Cover problem shows that ED remains NP-complete for 2P 3 -free (and thus, for P 7 -free) chordal graphs. In [1], it is shown that for P 6 -free chordal graphs, WED is solvable in polynomial time. Very recently, it has been shown by Lokshtanov et al [6] that ED is solvable in polynomial time for P 6 -free graphs in general; independently, in a direct approach, also Raffaele Mosca [9] It is well known that in a connected graph G with connected complement G, the maximal homogeneous sets are pairwise disjoint and can be determined in linear time using the so called modular decomposition (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Weighted Efficient Domination for $$P_6$$ -Free and for $$P_5$$ -Free Graphs

Brandstädt

Mosca

2016

Graph-Theoretic Concepts in Computer Science

Self Cite

View full text Add to dashboard Cite

In a finite undirected graph G = (V, E), a vertex v ∈ V dominates itself and its neighbors. A vertex set D ⊆ V in G is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete for P 7 -free graphs but solvable in polynomial time for P 5 -free graphs. Very recently, it has been shown by Lokshtanov et al. and independently by Mosca that ED is solvable in polynomial time for P 6 -free graphs.In this note, we show that, based on modular decomposition, ED is solvable in linear time for P 5 -free graphs.

show abstract

Weighted Efficient Domination for $P_6$-Free Graphs in Polynomial Time

Brandstädt¹,

Mosca²

2015

Preprint

View full text Add to dashboard Cite

In a finite undirected graph G = (V, E), a vertex v ∈ V dominates itself and its neighbors in G. A vertex set D ⊆ V is an efficient dominating set (e.d. for short) of G if every v ∈ V is dominated in G by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete for P 7 -free graphs but solvable in polynomial time for P 5 -free graphs. The P 6 -free case was the last open question for the complexity of ED on F -free graphs.Recently, Lokshtanov, Pilipczuk and van Leeuwen showed that weighted ED is solvable in polynomial time for P 6 -free graphs, based on their subexponential algorithm for the Maximum Weight Independent Set problem for P 6 -free graphs. Independently, at the same time, Mosca found a polynomial time algorithm for weighted ED on P 6 -free graphs using a direct approach. In this paper, we describe the details of this approach which is simpler and much faster, namely its time bound is O(n 6 m).

show abstract

Efficient Domination for Some Subclasses of $P_6$-Free Graphs in Polynomial Time

Cited by 3 publications

References 0 publications

On Efficient Domination for Some Classes of H-Free Chordal Graphs

On Efficient Domination for Some Classes of H-Free Chordal Graphs

Weighted Efficient Domination for $$P_6$$ -Free and for $$P_5$$ -Free Graphs

Weighted Efficient Domination for $P_6$-Free Graphs in Polynomial Time

Contact Info

Product

Resources

About