The Maker-Breaker Largest Connected Subgraph game

Given a graph G and k∈N, we introduce the following game played in G. Each round, Alice colours an uncoloured vertex of red, then Bob one blue (if any remain). Once every is coloured, wins if there connected red component order at least k, otherwise, wins. This Maker-Breaker version Largest Connected Subgraph introduced [Bensmail et al., The largest subgraph game, Algorithmica 84 (9) (2022) 2533–2555]. We want to compute cg(G), which maximum k such that G, regardless Bob's strategy. prove deciding whether cg(G)≥k PSPACE-complete, even bipartite, split, or planar graph. To better understand focus on A-perfect graphs, are graphs for cg(G)=⌈|V(G)|/2⌉, i.e., those can ensure connected. give sufficient conditions, terms minimum degrees number edges, be A-perfect. Also, show that, d≥4, arbitrarily large d-regular but no cubic with 18 Lastly, cg(G) computable linear time when P4-sparse (a superclass cographs).