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

Abstract

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).

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

“…Since (W)ED is NP-complete for 2P 3 -free graphs and polynomial for (P 5 + kP 2 )-free graphs [8,9], the class of P 6 -free graphs was the only open case. It was finally solved in [13,14] by a direct polynomial time approach (and in [27] by an indirect one).…”