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The Largest Connected Subgraph Game

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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, if Alice plays optimally, then 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 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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Notes

  1. Generalised Hex is a generalisation of Hex played on graphs, where the two players alternate picking vertices, with the first player aiming to join two fixed vertices s and t by a path formed by their vertices, while the second player aims at preventing this from happening.

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Authors and Affiliations

  1. Université Côte d’Azur, Inria, CNRS, I3S, Valbonne, France

    Julien Bensmail, Foivos Fioravantes & Nicolas Nisse

  2. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Fionn Mc Inerney

Authors
  1. Julien Bensmail
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  2. Foivos Fioravantes
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  3. Fionn Mc Inerney
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  4. Nicolas Nisse
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Correspondence to Fionn Mc Inerney.

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This work has been supported by the European Research Council (ERC) consolidator Grant No. 725978 SYSTEMATICGRAPH, the STIC-AmSud project GALOP, the PHC Xu Guangqi project DESPROGES, and the UCA\(^\textit{JEDI}\) Investments in the Future project managed by the National Research Agency (ANR-15-IDEX-01). An extended abstract of parts of this paper has been presented in [2]

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Bensmail, J., Fioravantes, F., Mc Inerney, F. et al. The Largest Connected Subgraph Game. Algorithmica 84, 2533–2555 (2022). https://doi.org/10.1007/s00453-022-00973-5

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  • DOI: https://doi.org/10.1007/s00453-022-00973-5

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