Dominating Cycles and Forbidden Pairs Containing $$P_{5}$$ P 5

Abstract

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 $$\mathcal {H}$$H of connected graphs, a graph G is said to be $$\mathcal {H}$$H-free if G does not contain any member of $$\mathcal {H}$$H as an induced subgraph. When $$|\mathcal {H}| = 2, \mathcal {H}$$|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 $$\{P_{5}, K_{4}^{-}\}$${P5,K4-}-free graph contains a longest cycle which is a dominating cycle. (ii) Every 2-connected $$\{P_{5}, W^{*}\}$${P5,Wź}-free graph contains a longest cycle which is a dominating cycle. Here $$P_{5}$$P5 is the path of order $$5, K_{4}^{-}$$5,K4- is the graph obtained from the complete graph of order 4 by removing one edge, and $$W^{*}$$Wź is the graph obtained from two triangles and an edge by identifying one vertex in each.