Weighted efficient domination for some classes of [formula omitted]-free and of [formula omitted]-free graphs

Abstract

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 claw-free graphs, for 2P3-free chordal graphs (and thus, for P7-free graphs), and for bipartite graphs. For the complexity of ED and its weighted version WED, a dichotomy for H-free graphs was reached: A graph H is called a linear forest if H is acyclic and claw-free, that is, if all its components are paths. Thus, the ED problem remains NP-complete for H-free graphs whenever H is not a linear forest. For every linear forest H, WED is either solvable in polynomial time or NP-complete for H-free graphs; the final result showed that WED is solvable in polynomial time for P6-free graphs.For (H1,H2)-free graphs, however, we are still far away from a dichotomy result. The main topics of this paper are: (1) to improve the time bounds and simplify the proofs (based on modular decomposition) for polynomial time cases of WED for some H-free graph classes; (2) to investigate the complexity of WED for some cases of (H1,H2)-free graphs such that WED is NP-complete for Hi-free graphs for at least one of i∈{1,2}. Since it is well known that WED is solvable in polynomial time for graph classes of bounded clique-width, we consider only classes of (H1,H2)-free graphs with unbounded clique-width.