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