Geodesic
A Classification of Cactus Graphs According to their Domination Number
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 613-626

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A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number, γ(G), of G is the minimum cardinality of a dominating set of G. The authors proved in [A new lower bound on the domination number of a graph, J. Comb. Optim. 38 (2019) 721–738] that if G is a connected graph of order n ≥ 2 with k ≥ 0 cycles and ℓ leaves, then γ(G) ≥ ⌈(n − ℓ + 2 − 2k)/3⌉. As a consequence of the above bound, γ(G) = (n − ℓ + 2(1 − k) + m)/3 for some integer m ≥ 0. In this paper, we characterize the class of cactus graphs achieving equality here, thereby providing a classification of all cactus graphs according to their domination number.
Keywords: domination number, lower bounds, cycles, cactus graphs 
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Hajian, Majid; Henning, Michael A.; Rad, Nader Jafari. A Classification of Cactus Graphs According to their Domination Number. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 613-626. http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a16/