Total Domination Versus Domination in Cubic Graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S has a neighbor in S. Further, if every vertex of G has a neighbor in S, then S is a total dominating set of G. The domination number, $$\gamma (G)$$γ(G), and total domination number, $$\gamma _{t}(G)$$γt(G), are the minimum cardinalities of a dominating set and total dominating set, respectively, in G. The upper domination number, $$\Gamma (G)$$Γ(G), and the upper total domination number, $$\Gamma _t(G)$$Γt(G), are the maximum cardinalities of a minimal dominating set and total dominating set, respectively, in G. It is known that $$\gamma _{t}(G)/\gamma (G) \le 2$$γt(G)/γ(G)≤2 and $$\Gamma _{t}(G)/\Gamma (G) \le 2$$Γt(G)/Γ(G)≤2 for all graphs G with no isolated vertex. In this paper we characterize the connected cubic graphs G satisfying $$\gamma _{t}(G)/\gamma (G) = 2$$γt(G)/γ(G)=2, and we characterize the connected cubic graphs G satisfying $$\Gamma _{t}(G)/\Gamma (G) = 2$$Γt(G)/Γ(G)=2.