On Total Domination and Minimum Maximal Matchings in Graphs

A subset $M$ of the edges of a graph $G$ is a matching if no two edges in $M$ are incident. A maximal matching is a matching that is not contained in a larger matching. A subset $S$ of vertices of a graph $G$ with no isolated vertices is a total dominating set of $G$ if every vertex of $G$ is adjacent to at least one vertex in $S$. Let $\mu^*(G)$ and $\gamma_t(G)$ be the minimum cardinalities of a maximal matching and a total dominating set in $G$, respectively. Let $\delta(G)$ denote the minimum degree in graph $G$. We observe that $\gamma_t(G)\leq 2\mu^*(G)$ when $1\leq \delta(G)\leq 2$ and $\gamma_t(G)\leq 2\mu^*(G)-\delta(G)+2$ when $\delta(G)\geq 3$. We show that the upper bound for the total domination number is tight for every fixed $\delta(G)$. We provide a constructive characterization of graphs $G$ satisfying $\gamma_t(G)= 2\mu^*(G)$ and a polynomial time procedure to determine whether $\gamma_t(G) = 2\mu^*(G)$ for a graph $G$ with minimum degree two.