Efficient Domination for Some Subclasses of P_6 -free Graphs in Polynomial Time
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چکیده
Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D 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 even for very restricted graph classes such as P7-free chordal graphs. The ED problem on a graph G can be reduced to the Maximum Weight Independent Set (MWIS) problem on the square of G. The complexity of the ED problem is an open question for P6-free graphs and was open even for the subclass of P6-free chordal graphs. In this paper, we show that squares of P6-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of (P6, house, hole, domino)-free graphs. This implies that ED/WeightedED is solvable in polynomial time for (P6, house, hole, domino)-free graphs; in particular, for P6-free chordal graphs. Moreover, based on our result that squares of P6-free graphs that have an e.d. are hole-free and some properties concerning odd antiholes, we show that squares of (P6, house)-free graphs ((P6, bull)-free graphs, respectively) that have an e.d. are perfect. This implies that ED/WeightedED is solvable in polynomial time for (P6, house)-free graphs and for (P6, bull)-free graphs (the time bound for (P6, house, hole, domino)-free graphs is better than that for (P6, house)-free graphs). The complexity of the ED problem for P6-free graphs remains an open question.
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Weighted Efficient Domination for $P_6$-Free Graphs in Polynomial Time
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 P7-free graphs but solvable in linear time for P5-fr...
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