Weighted efficient domination for some classes of H-free and of (H1,H2)-free graphs

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 2 P 3 -free chordal graphs (and thus, for P 7 -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 P 6 -free graphs. For ( H 1 , H 2 ) -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 ( H 1 , H 2 ) -free graphs such that WED is NP -complete for H i -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 ( H 1 , H 2 ) -free graphs with unbounded clique-width.