Domination game and minimal edge cuts

TL;DR

This paper explores the relationship between the domination game and minimal edge cuts in graphs, establishing bounds on the game domination number and introducing new graph classes to extend existing results.

Contribution

It establishes an upper bound on the game domination number based on minimal edge cuts and introduces Double-Staller graphs to demonstrate the bound's tightness.

Findings

Bound on game domination number using minimal edge cuts

Introduction of Double-Staller graphs for tightness demonstration

Extension of traceable graphs with known domination number bounds

Abstract

In this paper a relationship is established between the domination game and minimal edge cuts. It is proved that the game domination number of a connected graph can be bounded above in terms of the size of minimal edge cuts. In particular, if $C$ a minimum edge cut of a connected graph $G$, then $\gamma_g(G) \le \gamma_g(G\setminus C) + 2\kappa'(G)$. Double-Staller graphs are introduced in order to show that this upper bound can be attained for graphs with a bridge. The obtained results are used to extend the family of known traceable graphs whose game domination numbers are at most one-half their order. Along the way two technical lemmas, which seem to be generally applicable for the study of the domination game, are proved.