On Total Domination and Minimum Maximal Matchings in Graphs

Abstract

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 $μ^*(G)$ and $γ_t(G)$ be the minimum cardinalities of a maximal matching and a total dominating set in $G$, respectively. Let $δ(G)$ denote the minimum degree in graph $G$. We observe that $γ_t(G)\leq 2μ^*(G)$ when $1\leq δ(G)\leq 2$ and $γ_t(G)\leq 2μ^*(G)-δ(G)+2$ when $δ(G)\geq 3$. We show that the upper bound for the total domination number is tight for every fixed $δ(G)$. We provide a constructive characterization of graphs $G$ satisfying $γ_t(G)= 2μ^*(G)$ and a polynomial time procedure to determine whether $γ_t(G) = 2μ^*(G)$ for a graph $G$ with minimum degree two.