Relating the annihilation number and the total domination number of a tree

Authors:
Wyatt J. Desormeaux;Teresa W. Haynes;Michael A. Henning
Affiliations:
Department of Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa;Department of Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa and Department of Mathematics, East Tennessee State University, Johnson City, TN 37614-0002, USA;Department of Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa
Venue:
Discrete Applied Mathematics
Year:
2013

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Abstract

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number @c"t(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n=2. We show that @c"t(T)@?a(T)+1, and we characterize the extremal trees achieving equality in this bound.