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The Closed Hull Game and the Closed Interval Game

Authors Samuel N. Araújo , Fabrício Benevides , Nicolas Martins, Nicolas Nisse , Rudini Sampaio

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LIPIcs.FUN.2026.4.pdf
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ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Combinatorial games in graphs
  • graph convexity
  • PSPACE

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Abstract

Given a set S of vertices in a graph G, its geodesic interval is the set I(S) containing S and all vertices on a shortest path between vertices of S. A set S is convex if I(S) = S. Moreover, the convex hull ℋ(S) of S is the smallest convex set containing S. In 1984, Harary introduced convexity games where two players, Alice and Bob, alternately select vertices of a graph G = (V,E) such that, if the set of already selected vertices is S, the next player can only select a vertex in V ⧵ I(S) (closed interval game) or in V ⧵ ℋ(S) (closed hull game). Normal and misère versions of these games have been studied and here, we introduced the optimization variants of them. Formally, given a graph G and k ∈ ℕ, Alice wins if the game ends after at most k vertices have been selected and Bob wins otherwise. The corresponding problem consists of determining which player has a winning strategy.
We prove that the closed interval optimization game is PSPACE-complete in graphs with diameter 4 and that the closed hull optimization game is NP-hard in bipartite graphs and in split graphs. On the positive side, we prove that both games can be solved in polynomial time in trees and that the closed hull optimization game can be solved in polynomial time in cobipartite graphs. We conjecture that the closed interval optimization game is NP-hard in cobipartite graphs and that the closed hull optimization game is PSPACE-complete in general graphs.

Cite As Get BibTex

Samuel N. Araújo, Fabrício Benevides, Nicolas Martins, Nicolas Nisse, and Rudini Sampaio. The Closed Hull Game and the Closed Interval Game. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.4

Author Details

Samuel N. Araújo
  • Université Côte d’Azur, Inria, CNRS, I3S, Sophia Antipolis, France
  • PARGO Research Group, Universidade Federal do Ceará, Fortaleza, Brazil
  • Inst. Fed. Educação, Ciência e Tecnologia do Ceará, IFCE, Crato, Brazil
Fabrício Benevides
  • PARGO Research Group, Universidade Federal do Ceará, Fortaleza, Brazil
Nicolas Martins
  • Inst. de Eng. e Desenvolvimento Sustentável, UNILAB, Redenção, Brazil
Nicolas Nisse
  • Université Côte d’Azur, Inria, CNRS, I3S, Sophia Antipolis, France
Rudini Sampaio
  • PARGO Research Group, Universidade Federal do Ceará, Fortaleza, Brazil

Funding

Research supported by the CAPES-Cofecub project Ma 1004/23, by the Inria Associated Team CANOE, by the research grants ANR P-GASE and by the French government, through the EUR DS4H Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-17-EURE-0004.

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