Well-dominated graphs without cycles of lengths 4 and 5

Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph $G$ is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that recognizing well-dominated graphs can be done polynomially for graphs without cycles of lengths $4$ and $5$, by proving that a graph belonging to this family is well-dominated if and only if it is well-covered. Assume that a weight function $w$ is defined on the vertices of $G$. Then $G$ is $w$-well-dominated} if all its minimal dominating sets are of the same weight. We prove that the set of weight functions $w$ such that $G$ is $w$-well-dominated is a vector space, and denote that vector space by $WWD(G)$. We prove that $WWD(G)$ is a subspace of $WCW(G)$, the vector space of weight functions $w$ such that $G$ is $w$-well-covered. We provide a polynomial characterization of $WWD(G)$ for the case that $G$ does not contain cycles of lengths $4$, $5$, and $6$.