Domination and total domination in cubic graphs of large girth

Abstract

The domination number γ(G) and the total domination number γ t (G) of a graph G without an isolated vertex are among the most well studied parameters in graph theory. While the inequality γ t (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 to improve the inequality by much.In the present paper we consider the relation of γ(G) and γ t (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 γ t (G), which depends on the value of γ(G). As a consequence, we can considerably improve the inequality γ t (G) ≤ 2γ(G) for cubic graphs of large girth.

Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

“…Note that similar upper bounds involving the girth and other parameters of the graph can be found in many papers, e.g. in [10,12,16,17], while results for plane triangulations and maximal outerplanar graphs were established in [13] and [8].…”