  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.
