Realizations of the game domination number

Domination game is a game on a finite graph which includes two players. First player, Dominator, tries to dominate a graph in as few moves as possible; meanwhile the second player, Staller, tries to hold him back and delay the end of the game as long as she can. In each move at least one additional vertex has to be dominated. The number of all moves in the game in which Dominator makes the first move and both players play optimally is called the game domination number and is denoted by . The total number of moves in a Staller-start game is denoted by . It is known that for any graph . Graph realizes a pair if and . It is shown that pairs for all can be realized by a family of 2-connected graphs. We also present 2-connected classes which realize pairs and . Exact game domination number for combs and 1-connected realization of the pair (2k + 1, 2k) are also given.