Domination and total domination in cubic graphs of large girth

Abstract

The domination number γ(G) and the total domination number γt(G) of a graph G without an isolated vertex are among the most well-studied parameters in graph theory. While the inequality γt(G)≤2γ(G) is an almost immediate consequence of the definition, the extremal graphs for this inequality are not well understood. Furthermore, even very strong additional assumptions do not allow us to improve the inequality by much.In the present paper we consider the relation of γ(G) and γt(G) for cubic graphs G of large girth. Clearly, in this case γ(G) is at least n(G)/4 where n(G) is the order of G. If γ(G) is close to n(G)/4, then this forces a certain structure within G. We exploit this structure and prove an upper bound on γt(G), which depends on the value of γ(G). As a consequence, we can considerably improve the inequality γt(G)≤2γ(G) for cubic graphs of large girth.