A Classification of Cactus Graphs According to Their Total Domination Number

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 >= 2 with k >= 0 cycles and l leaves. Recently, the authors have proved that gamma(t) (G) >= 1/2 (n - l + 2) - k. As a consequence of this bound, gamma(t) (G) = 1/2 (n - l + 2 + m) - k for some integer m >= 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.