Structure of squares and efficient domination in graph classes

For any two vertices u and v in a graph G, dG(u,v) denote the distance between u and v in G. The square of a graph G=(V,E) is the graph G2=(V,E2) such that uv∈E2 if and only if dG(u,v)∈{1,2}. An efficient dominating set (e.d. for short) in G is a subset D of vertices such that D is an independent set and each vertex outside D has exactly one neighbor in D. In general, (i) if P is a (hereditary) graph property, and if a graph G satisfies P, then G2 does not need to satisfy P, and (ii) if H is any graph, and if G is a H-free graph that has an e.d., then G2 need not be H-free. In this paper, we show that the squares of (P6, banner)-free graphs that have an e.d. are again (P6, banner)-free. This result together with some known results implies that the Efficient Dominating Set problem (which asks for the existence of an e.d. in a given graph G) can be solved in time O(n3) for (P6, banner)-free graphs. Here, Pt denotes the chordless path on t vertices, and a banner is the graph obtained from a chordless cycle on four vertices by adding a vertex that has exactly one neighbor on the cycle.