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Impartial geodetic building games on graphs

Author

Listed:
  • Bret J. Benesh

    (Saint John’s University)

  • Dana C. Ernst

    (Northern Arizona University)

  • Marie Meyer

    (Lewis University)

  • Sarah K. Salmon

    (University of Colorado Boulder Campus Box 395)

  • Nándor Sieben

    (Northern Arizona University)

Abstract

A subset of the vertex set of a graph is geodetically convex if it contains every vertex on any shortest path between two elements of the set. The convex hull of a set of vertices is the smallest convex set containing the set. We study variations of two games introduced by Buckley and Harary, where two players take turns selecting previously-unselected vertices of a graph until the convex hull of the jointly-selected vertices becomes too large. The last player to move is the winner. The achievement game ends when the convex hull contains every vertex. In the avoidance game, the convex hull is not allowed to contain every vertex. We determine the nim-value of these games for several graph families.

Suggested Citation

  • Bret J. Benesh & Dana C. Ernst & Marie Meyer & Sarah K. Salmon & Nándor Sieben, 2024. "Impartial geodetic building games on graphs," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(4), pages 1335-1368, December.
    • Handle: RePEc:spr:jogath:v:53:y:2024:i:4:d:10.1007_s00182-024-00916-0
    DOI: 10.1007/s00182-024-00916-0
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