Total dominating functions of graphs: antiregularity versus regularity

A set S of vertices in a nontrivial connected graph G is a total dominating set if every vertex of G is adjacent to some vertex of S. The minimum cardinality of a total dominating set for G is the total domination number of G. A function h : V (G) → {0, 1} is a total dominating function of a graph G if σh(v) = ∑ u∈N(v) h(u) ≥ 1 for every vertex v of G. A total dominating function h of a nontrivial graph G is irregular if σh(u) 6= σh(v) for every two vertices u and v of G. While no graph possesses an irregular total dominating function, a graph G has an antiregular total dominating function h if there are exactly two vertices u and v of G such that σh(u) = σh(v). It is shown that for every integer n ≥ 3, there are exactly two non-isomorphic graphs of order n having an antiregular total dominating function. If h is a total dominating function of a graph G such that σh(v) is the same constant k for every vertex v of G, then h is a k-regular total dominating function of G. We present some results dealing with properties of regular total dominating functions of graphs. In particular, regular total dominating functions of trees are investigated. The only possible regular total dominating functions for a nontrivial tree are 1-regular total dominating functions. We characterize those trees having a 1-regular total dominating function. We also investigate k-regular total dominating functions of several well-known classes of regular graphs for various values of k.