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The Maker-Breaker Largest Connected Subgraph Game

Julien Bensmail 1 Foivos Fioravantes 1 Fionn Mc Inerney 2 Nicolas Nisse 1 Nacim Oijid 3
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
3 GOAL - Graphes, AlgOrithmes et AppLications
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : Given a graph $G$ and an integer $k \in \mathbb{N}$, we introduce the following Maker-Breaker game played in $G$. Each round, first Alice colours an uncoloured vertex of $G$ red, and then Bob colours an uncoloured vertex blue (if any remain). Once all the vertices have been coloured, Alice wins if there exists a connected red component of order at least $k$, and if not, then Bob wins. This game is a Maker-Breaker version of the Largest Connected Subgraph game introduced in [Bensmail, Fioravantes, Mc Inerney, and Nisse, WG 2021]. We are interested in computing $c_g(G)$, which is the maximum $k$ such that Alice wins in $G$, regardless of what Bob does. Given a graph $G$ and an integer $k\geq 1$, we prove that deciding whether $c_g(G)\geq k$ is PSPACE-complete, even if $G$ is restricted to be in the class of bipartite graphs of diameter~$4$, split graphs, or planar graphs. Then, we focus on {\it A-perfect} graphs, namely, graphs for which $c_g(G)=\left \lceil \frac{|V(G)|}{2}\right \rceil$, {\it i.e.}, the maximum possible value. We show that there exist arbitrarily large A-perfect $d$-regular graphs for any $d \geq 4$, but, surprisingly, that no cubic graph with order at least $133$ is A-perfect. Moreover, we give sufficient conditions, in terms of the number of edges or the maximum and minimum degree, for a graph to be A-perfect. Finally, we show that $c_g(G)$ can be computed in polynomial time when $G$ is a $P_4$-sparse graph (a superclass of cographs). We conclude with many open questions. In particular, natural graph classes such as trees and grid-like graphs seem to be difficult to deal with. We only give partial results in the case of some grid-like graphs.
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https://hal.inria.fr/hal-03500888
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Submitted on : Wednesday, December 22, 2021 - 5:08:28 PM
Last modification on : Tuesday, January 4, 2022 - 6:33:19 AM

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Julien Bensmail, Foivos Fioravantes, Fionn Mc Inerney, Nicolas Nisse, Nacim Oijid. The Maker-Breaker Largest Connected Subgraph Game. [Research Report] Université Côte d’Azur, CNRS, Inria, I3S, Biot, France. 2021. ⟨hal-03500888⟩

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