On Maker-Breaker domination game critical graphs
Bo\v{s}tjan Bre\v{s}ar, Tanja Dravec, Kirsti Kuenzel, Douglas F. Rall

TL;DR
This paper investigates the structure of critical graphs in the Maker-Breaker domination game, specifically those with a game invariant of 2, providing characterizations for graphs with cut-vertices and identifying unique non-bipartite, triangle-free cases.
Contribution
The paper characterizes 2-$oldsymbol{ extgamma_{MB}'}$-critical graphs with cut-vertices and proves the uniqueness of $C_5$ as the only non-bipartite, triangle-free critical graph.
Findings
Characterized 2-$ extgamma_{MB}'$-critical graphs with cut-vertices.
Identified two infinite families of such graphs.
Proved $C_5$ is the only non-bipartite, triangle-free 2-$ extgamma_{MB}'$-critical graph.
Abstract
The Maker-Breaker domination game is played on a graph by Dominator and Staller who alternate turns selecting an unplayed vertex of . 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 is the number of Dominator's moves in the game played on 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 --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 with and for every edge in . The authors characterized bipartite --critical…
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Taxonomy
TopicsDistributed and Parallel Computing Systems · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
