A Classification of Cactus Graphs According to Their Total Domination Number

Abstract

A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The total domination number, $\gamma _t(G)$, is the minimum cardinality of a total dominating set of G. A cactus is a connected graph in which every edge belongs to at most one cycle. Equivalently, a cactus is a connected graph in which every block is an edge or a cycle. Let G be a connected graph of order $n \ge 2$ with $k \ge 0$ cycles and $\ell$ leaves. Recently, the authors have proved that $\gamma _t(G) \ge \frac{1}{2}(n-\ell +2) - k$. As a consequence of this bound, $\gamma _t(G) = \frac{1}{2}(n-\ell +2+m) - k$ for some integer $m \ge 0$. In this paper, we characterize the class of cactus graphs achieving equality in this bound, thereby providing a classification of all cactus graphs according to their total domination number.