On total dominating sets in graphs

A set S of vertices in a graph G(V,E) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The domination number of a graph G denoted by γ(G) is the minimum cardinality of a dominating set in G. Respectively the total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. An upper bound for γt(G) which has been achieved by Cockayne and et al. in [1] is: for any graph G with no isolated vertex and maximum degree ∆(G) and n vertices, γt(G) ≤ n−∆(G) + 1. Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for γt(G). Also, for circular complete graphs, we determine the value of γt(G). 2000 Mathematics Subject Classification: 05c69