Domination game: Effect of edge- and vertex-removal

The domination game is played on a graph G by two players, named Dominator and Staller. They alternatively select vertices of G such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator’s goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Game 2 are variants of the game in which Dominator and Staller has the first move, respectively. The game domination number γ g ( G ) and the Staller-start game domination number γ ′ ( G ) are the number of vertices chosen in Game 1 and Game 2, respectively. It is proved that if e ∈ E ( G ) , then | γ g ( G ) − γ g ( G − e ) | ≤ 2 and | γ ′ ( G ) − γ ′ ( G − e ) | ≤ 2 , and that each of the possibilities here is realizable by connected graphs G for all values of γ g ( G ) and γ ′ ( G ) larger than 5. For the remaining small values it is either proved that realizations are not possible or realizing examples are provided. It is also proved that if v ∈ V ( G ) , then γ g ( G ) − γ g ( G − v ) ≤ 2 and γ ′ ( G ) − γ ′ ( G − v ) ≤ 2 . Possibilities here are again realizable by connected graphs G in almost all the cases, the exceptional values are treated similarly as in the edge-removal case.