Relating ordinary and total domination in cubic graphs of large girth

Abstract

Abstract. For a cubic graph G of order n, girth at least g, and domination number 1 4 + n for some ≥ 0, we show that the total domination number of G is at most 13 32For a finite, simple, andSimilarly, a set T of vertices of G is a total dominating set of G if every vertex in V (G) has a neighbor in T . Note that a graph has a total dominating set exactly if it has no isolated vertex. The minimum cardinalities of a dominating and a total dominating set of G are known as the domination number γ (G) of G and the total domination number γ t (G) of G, respectively. These two parameters are among the most fundamental and well studied parameters in graph theory [3,4,6]. In view of their computational hardness especially upper bounds were investigated in great detail. The two parameters are related by some very simple inequalities. Let G be a graph without isolated vertices. Since every total dominating set of G is also a dominating set of G, we immediately obtainSimilarly, if D is a dominating set of G, then adding, for every isolated vertex u of the subgraph G [D] of G induced by D, a neighbor of u in G