Bounds relating the weakly connected domination number to the total domination number and the matching number

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Abstract

Let G = (V, E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V, E ∩ (S × V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted γwc (G). A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt (G) of G. In this paper, we show that frac(1, 2) (γt (G) + 1) ≤ γwc (G) ≤ frac(3, 2) γt (G) - 1. Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α′ (G). We also establish that γwc (G) ≤ α′ (G), and show that γwc (T) = α′ (T) for every tree T. © 2009 Elsevier B.V. All rights reserved.

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Hattingh, J. H., & Henning, M. A. (2009). Bounds relating the weakly connected domination number to the total domination number and the matching number. Discrete Applied Mathematics, 157(14), 3086–3093. https://doi.org/10.1016/j.dam.2009.06.008

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