# A characterization of trees having a minimum vertex cover which is also a minimum total dominating set

## TL;DR

This paper provides a constructive characterization of trees that possess a vertex cover which is also a minimum total dominating set, linking two important graph concepts.

## Contribution

It introduces a new characterization of trees with a $( au_t- au)$-set, connecting minimum vertex covers and total domination in trees.

## Key findings

- Identifies trees with a $( au_t- au)$-set.
- Provides a constructive method for characterization.
- Bridges concepts of vertex cover and total domination.

## Abstract

A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no isolated vertices. A $(\gamma_t-\tau)$-set of $G$ is a minimum vertex cover which is also a minimum total dominating set. In this article we give a constructive characterization of trees having a $(\gamma_t-\tau)$-set.