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 $6$.Comment: 10 pages, 2 figure

Well-Totally-Dominated Graphs

A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions.Comment: 13 pages, 2 figure

On well-dominated graphs

A graph is \emph{well-dominated} if all of its minimal dominating sets have the same cardinality. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order $2$. Under the assumption that at least one of the connected graphs $G$ or $H$ has no isolatable vertices, we prove that the direct product of $G$ and $H$ is well-dominated if and only if either $G=H=K_3$ or $G=K_2$ and $H$ is either the $4$-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most $2$.Comment: 16 pages, 2 figure