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 γMB'(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 $2$-γMB'-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 γMB'(G)=2 and γMB'(G-e)>2 for every edge e in G. The authors characterized bipartite $2$-γMB'-critical graphs, and found an example of a non-bipartite $2$-γMB'-critical graph. In this paper, we characterize the $2$-γMB'-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 $2$-γMB'-critical graph.