# The Largest Connected Subgraph Game

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : This paper introduces the largest connected subgraph game played on an undirected graph $G$. In each round, Alice first colours an uncoloured vertex of $G$ red, and then, Bob colours an uncoloured vertex of $G$ blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than the order of any blue (red, resp.) connected subgraph. We first prove that Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or has at least $11$ vertices, and that Alice wins in cycles if and if only if the cycle is of odd length. Lastly, we give an algorithm to determine the outcome of the game in cographs in linear time.
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https://hal.inria.fr/hal-03137305
Contributor : Fionn Mc Inerney <>
Submitted on : Friday, March 5, 2021 - 9:16:50 PM
Last modification on : Tuesday, March 9, 2021 - 3:27:24 AM

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• HAL Id : hal-03137305, version 3

### Citation

Julien Bensmail, Foivos Fioravantes, Fionn Mc Inerney, Nicolas Nisse. The Largest Connected Subgraph Game. [Research Report] Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France; CISPA Helmholtz Center for Information Security, Saarbrücken, Germany. 2021. ⟨hal-03137305v3⟩

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