Improved bounds on the domination number of a tree

• Published in: Discrete Applied Mathematics Vol. 177; pp. 88 - 94
• Main Authors: Desormeaux, Wyatt J., Haynes, Teresa W., Henning, Michael A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 20-11-2014
• Subjects: Degree sequence; Domination; Slater number; Trees; Domination; Trees; Slater number; Degree sequence
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2014.05.037

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The Slater number sℓ(G) is the smallest integer t such that t added to the sum of the first t terms of the non-increasing degree sequence of G is at least as large as the number of vertices of G. It is well-known that γ(G)≥sℓ(G). If G has n vertices with minimum degree  δ≥1 and maximum degree  Δ, then we show that ⌈n/(Δ+1)⌉≤sℓ(G)≤⌈n/(δ+1)⌉. Let T be a tree on n≥3 vertices with n1 vertices of degree  1. We prove that γ(T)≤3sℓ(T)−2, and we characterize the trees that achieve equality in this bound. Lemanska (2004) proved that γ(T)≥(n−n1+2)/3. We improve this result by showing that sℓ(T)≥(n−n1+2)/3 and establishing the existence of trees T for which the difference between the Slater number sℓ(T) and (n−n1+2)/3 is arbitrarily large. Further, the trees T satisfying sℓ(T)=(n−n1+2)/3 are characterized.