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, γt(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _t(G)$$\end{document}, 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\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 2$$\end{document} with k≥0\documentclass[12pt]{minimal}
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\begin{document}$$k \ge 0$$\end{document} cycles and ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} leaves. Recently, the authors have proved that γt(G)≥12(n-ℓ+2)-k\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _t(G) \ge \frac{1}{2}(n-\ell +2) - k$$\end{document}. As a consequence of this bound, γt(G)=12(n-ℓ+2+m)-k\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _t(G) = \frac{1}{2}(n-\ell +2+m) - k$$\end{document} for some integer m≥0\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 0$$\end{document}. 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.
