# On Total Domination and Minimum Maximal Matchings in Graphs

## TL;DR

This paper investigates the relationship between total domination and maximum minimal matchings in graphs, establishing tight bounds and providing characterizations and algorithms for specific cases based on minimum degree.

## Contribution

It introduces new bounds relating total domination number and maximum minimal matching size, and offers a characterization and polynomial-time algorithm for graphs with minimum degree two.

## Key findings

- Established bounds: (G)(G)(G) (G) (G) when (G)(G)(G) (G) (G)
- Proved the bounds are tight for every fixed minimum degree
- Provided a constructive characterization and polynomial-time decision procedure for graphs with minimum degree two

## 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 $\mu^*(G)$ 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.