On αα-total domination in graphs

Let G=(V,E)G=(V,E) be a graph with no isolated vertex. A subset of vertices SS is a total dominating set if every vertex of GG is adjacent to some vertex of SS. For some αα with 0<α≤10<α≤1, a total dominating set SS in GG is an αα-total dominating set if for every vertex v∈V∖Sv∈V∖S, |N(v)∩S|≥α|N(v)||N(v)∩S|≥α|N(v)|. The minimum cardinality of an αα-total dominating set of GG is called the αα-total domination number of GG. In this paper, we study αα-total domination in graphs. We obtain several results and bounds for the αα-total domination number of a graph GG.