New polynomial cases of the weighted efficient domination problem

Abstract

Let G be a finite undirected graph. A vertex dominates itself and its neighbors in G. A vertex set D is an efficient dominating set (e.d. 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. in G, is known to be ℕℙ-complete even for very restricted graph classes. In particular, the ED problem remains -complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of P7-free graphs, including (P2 + P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our ℕℙ-completeness result for the ED problem on planar bipartite graphs with maximum degree 3. © 2013 Springer-Verlag.