On Maker-Breaker domination game critical graphs

The Maker-Breaker domination game is played on a graph G by Dominator and Staller who alternate turns selecting an unplayed vertex of G. The goal of Dominator is that the vertices he selected during the game form a dominating set while Staller's goal is to prevent this from happening. The graph invariant γ ₌₁' (G) is the number of Dominator's moves in the game played on G in which he can achieve his goal when Staller makes the first move and both players play optimally. In this paper, we continue the investigation of 2-γ ₌₁'-critical graphs, initiated in Divarakan et al. , Maker--Breaker domination game critical graphs, Discrete Appl. \ Math. 368 (2025) 126--134, which are defined as the graphs G with γ ₌₁' (G) =2 and γ ₌₁' (G-e) >2 for every edge e in G. The authors characterized bipartite 2-γ ₌₁'-critical graphs, and found an example of a non-bipartite 2-γ ₌₁'-critical graph. In this paper, we characterize the 2-γ ₌₁'-critical graphs that have a cut-vertex, which are represented by two infinite families. In addition, we prove that C₅ is the only non-bipartite, triangle-free 2-γ ₌₁'-critical graph.