The Maker-Breaker Largest Connected Subgraph game

Julien Bensmail, Foivos Fioravantes, Fionn Mc Inerney, Nicolas Nisse, Nacim Oijid

Abstract: Given a graph G and k∈N<math><mi is="true">k</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></math>, 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)<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">g</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></math>, which is the maximum k such that Alice wins in G, regardless of Bob's strategy.Given a graph G and k∈N<math><mi is="true">k</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></math>, we prove that deciding whether cg(G)≥k<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">g</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">≥</mo><mi is="true">k</mi></math> 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⌉<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">g</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mo stretchy="false" is="true">⌈</mo><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">|</mo><mo stretchy="false" is="true">/</mo><mn is="true">2</mn><mo stretchy="false" is="true">⌉</mo></math>, 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<math><mi is="true">d</mi><mo is="true">≥</mo><mn is="true">4</mn></math>, 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)<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">g</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></math> is computable in linear time when G is a P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math>-sparse graph (a superclass of cographs).