Weighted efficient domination for some classes of [Formula presented]-free and of [Formula presented]-free graphs

Abstract

A vertex set [Formula presented] in a finite undirected graph [Formula presented] is an efficient dominating set (e.d.s. for short) of [Formula presented] if every vertex of [Formula presented] is dominated by exactly one vertex of [Formula presented]. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in [Formula presented], is known to be [Formula presented]-complete even for very restricted graph classes such as for claw-free graphs, for [Formula presented]-free chordal graphs (and thus, for [Formula presented]-free graphs), and for bipartite graphs. For the complexity of ED and its weighted version WED, a dichotomy for [Formula presented]-free graphs was reached: A graph [Formula presented] is called a linear forest if [Formula presented] is acyclic and claw-free, that is, if all its components are paths. Thus, the ED problem remains [Formula presented]-complete for [Formula presented]-free graphs whenever [Formula presented] is not a linear forest. For every linear forest [Formula presented], WED is either solvable in polynomial time or [Formula presented]-complete for [Formula presented]-free graphs; the final result showed that WED is solvable in polynomial time for [Formula presented]-free graphs. For [Formula presented]-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 [Formula presented]-free graph classes; (2) to investigate the complexity of WED for some cases of [Formula presented]-free graphs such that WED is [Formula presented]-complete for [Formula presented]-free graphs for at least one of [Formula presented]. Since it is well known that WED is solvable in polynomial time for graph classes of bounded clique-width, we consider only classes of [Formula presented]-free graphs with unbounded clique-width.