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 α with 0<α≤1, a total dominating set S in G is an α-total dominating set if for every vertex v∈V\S, |N(v)∩S|≥α|N(v)|. The minimum cardinality of an α-total dominating set of G is called the α-total domination number of G. In this paper, we study α-total domination in graphs. We obtain several results and bounds for the α-total domination number of a graph G.