Dominating cycles and forbidden pairs containing a path of order 5

A cycle is a graph is dominating if every edge of the graph is incident with a vertex of the cycle. 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 $\{P_{5}, K_{4}^{-}\}$-free graph contains a longest cycle which is a dominating cycle. (ii) Every $2$-connected $\{P_{5}, W^{*}\}$-free graph contains a longest cycle which is a dominating cycle. Here $P_{5}$ 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 a graph obtained from two triangles and an edge by identifying one vertex in each.
Comment: 17pages, 7 figures