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, γ_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 γt(G)≥12(n-ℓ+2)-k. As a consequence of this bound, γt(G)=12(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.