Effect of an edge subdivision on game domination numbers

Abstract

Domination game is a game on a graph [Formula: see text] played by two players called Dominator and Staller. They alternately choose vertices of [Formula: see text] such that each move dominates at least one new undominated vertex. The game ends when all vertices are dominated. Dominator’s goal is to finish the game as soon as possible, while Staller’s goal is to prolong it as much as she can. When Dominator moves first, the game is a Dominator-start game; when Staller moves first, it is a Staller-start game. The game domination numbers [Formula: see text] and [Formula: see text] are the sizes of the final dominating sets when both players play optimally for the Dominator-start game and for the Staller-start game, respectively. For a graph [Formula: see text] and an edge [Formula: see text], we show that [Formula: see text] and [Formula: see text], where [Formula: see text] is is the graph resulting from subdividing the edge [Formula: see text] in [Formula: see text]. We also demonstrate that each difference satisfying the above bounds are realizable by infinitely many connected graphs.