On efficient domination for some classes of [formula omitted]-free bipartite 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 H-free graph classes such as for 2P3-free chordal graphs while it is solvable in polynomial time for P6-free graphs. Here we focus on bipartite graphs: We show that (weighted) ED can be solved in polynomial time for H-free bipartite graphs when H is P7 or ℓP4 for fixed ℓ, and similarly for P9-free bipartite graphs with vertex degree at most 3, and when H is S2,2,4. Moreover, we show that ED is NP-complete for bipartite graphs with diameter at most 6.