Mycielskian of graphs with small game domination number

Abstract

Domination game is a game played on a finite, undirected graph G, between two players Dominator and Staller. During the game, the players alternately choose vertices of G such that each chosen vertex dominates at least one new vertex that is not dominated by previously chosen vertices. The aim of Dominator is to finish the game as early as possible while that of Staller is to delay the process as much as possible. The game domination number γg(G) is the total number of moves in the game when Dominator starts and both players play optimally. Similarly the staller start game domination number γg′(G) is the total number of moves in the game when Staller starts and both players play optimally. Here the domination game in the Mycielskian of a graph is studied. We establish bounds for the game domination number of Mycielskian of a graph in terms of its domination number and also in terms of its game domination number. The Mycielskian of a graph with small game domination number are characterised.