Structure of squares and efficient domination in graph classes

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.