1. A new lower bound on the total domination number of a graph.
- Author
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Hajian, Majid, Henning, Michael A., and Rad, Nader Jafari
- Subjects
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DOMINATING set , *GRAPH connectivity , *MATHEMATICS - 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, γt(G), is the minimum cardinality of a total dominating set of G. Chellali and Haynes [J. Combin. Math. Combin. Comput.58 (2006), 189–193] showed that if T is a nontrivial tree of order n, with ℓ leaves, then γt(T) ≥ (n − ℓ + 2)/2. In this paper, we first characterize all trees T of order n with ℓ leaves satisfying γt(T) = ⌈(n−ℓ+2)/2⌉. We then generalize this result to connected graphs and show that if G is a connected graph of order n ≥ 2 with k ≥ 0 cycles and ℓ leaves, then γt(G) ≥ ⌈(n − ℓ + 2)/2⌉ − k. We also characterize the graphs G achieving equality for this new bound. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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