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On the Complexity of the Maker-Breaker Happy Vertex Game

Authors Mathieu Hilaire, Perig Montfort, Nacim Oijid

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LIPIcs.FUN.2026.24.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
Keywords
  • Maker-Breaker game
  • Domination game
  • happy vertex game
  • scoring game
  • complexity

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Abstract

Given a c-colored graph G, a vertex v of G is said to be happy if it has the same color as all its neighbors. The notion of happy vertices was introduced by Zhang and Li [Peng Zhang and Angsheng Li, 2015] to compute the homophily of a graph. Eto, Fujimoto, Kiya, Matsushita, Miyano, Murao and Saitoh [Hiroshi Eto et al., 2025] introduced the Maker-Maker version of the Happy vertex game, where two players compete to claim more happy vertices than their opponent. We introduce here the Maker-Breaker happy vertex game: two players, Maker and Breaker, alternately color the vertices of a graph with their respective colors. Maker aims to maximize the number of happy vertices at the end, while Breaker aims to prevent her. This game is also a scoring version of the Maker-Breaker domination game introduced by Duchene, Gledel, Parreau and Renault [Duchene et al., 2020], as a happy vertex corresponds exactly to a vertex that is not dominated in the domination game. Therefore, this game is a very natural game on graphs and can be studied within the scope of scoring positional games [Bagan et al., 2024]. We initiate here the complexity study of this game, by proving that computing its score is PSPACE-complete on trees, NP-hard on caterpillars, and polynomial on subdivided stars. Finally, we provide the exact value of the score on graphs of maximum degree 2, and we provide an FPT-algorithm to compute the score on graphs of bounded neighborhood diversity. An important contribution of the paper is that, to achieve our hardness results, we introduce a new type of incidence graph called the literal-clause incidence graph for 2-SAT formulas. We prove that QMAX 2-SAT remains PSPACE-complete even if this graph is acyclic, and that MAX 2-SAT remains NP-complete, even if this graph is acyclic and has maximum degree 2, i.e. is a union of paths. We demonstrate the importance of this contribution by proving that Incidence, the scoring positional game played on a graph is also PSPACE-complete when restricted to forests.

Cite As Get BibTex

Mathieu Hilaire, Perig Montfort, and Nacim Oijid. On the Complexity of the Maker-Breaker Happy Vertex Game. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.24

Author Details

Mathieu Hilaire
  • Univ. Bordeaux, Bordeaux INP, LaBRI UMR CNRS 5800, F-33400, Talence, France
Perig Montfort
  • ENS de Lyon, France
Nacim Oijid
  • Umeå University, Sweden

Funding

This research was supported by the ANR project P-GASE (ANR-21-CE48-0001-01).
  • Oijid, Nacim: Kempe Foundation Grant No. JCSMK24-515 (Sweden).

References

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