Dominating Cycles and Forbidden Pairs Containing P5

A cycle C in a graph G is dominating if every edge of G is incident with at least one vertex of C. For a set H of connected graphs, a graph G is said to be H-free if G does not contain any member of H as an induced subgraph. When |H| = 2, H is called a forbidden pair. In this paper, we investigate the characterization of the class of the forbidden pairs guaranteeing the existence of a dominating cycle and show the following two results: (i) Every 2-connected {P5,K 4 }-free graph contains a longest cycle which is a dominating cycle. (ii) Every 2-connected {P5,W ∗}-free graph contains a longest cycle which is a dominating cycle. Here P5 is the path of order 5, K− 4 is the graph obtained from the complete graph of order 4 by removing one edge, and W ∗ is the graph obtained from two triangles and an edge by identifying one vertex in each.