A note on α-total domination in cubic graphs

Let G = (V, E) be a graph with no isolated vertex. A subset of vertices S is a total dominating set if every vertex of G is adjacent to some vertex of S. For some alpha with 0 < alpha <= 1, a total dominating set S in G is an alpha-total dominating set if for every vertex v is an element of V \ S, vertical bar N(v) boolean AND S vertical bar >= alpha vertical bar N(v)vertical bar. The alpha-total domination number of G, denoted by gamma(alpha t) (G), is the minimum cardinality of an alpha-total dominating set of G. In Henning and Rad (2042), Henning and Rad posed the following question: Let G be a connected cubic graph with order n. Is it true that gamma(alpha t) (G) <= 3n/4 for 2/3 < alpha <= 1 ? In this paper, we give a positive answer toward this question. Furthermore, we give a characterization on cubic graphs attaining the bound for the alpha-total domination number. (C) 2016 Elsevier B.V. All rights reserved.