On α-total domination in 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 nu is an element of V \ S, vertical bar N(upsilon) boolean AND S vertical bar >= alpha vertical bar N(upsilon)vertical bar. The minimum cardinality of an alpha-total dominating set of G is called the alpha-total domination number of G. In this paper, we study alpha-total domination in graphs. We obtain several results and bounds for the alpha-total domination number of a graph G. (C) 2011 Elsevier B.V. All rights reserved.