Impartial geodetic building games on graphs

TL;DR

This paper explores two vertex-selection games on graphs involving geodetic convexity, analyzing their strategies and outcomes for various graph types, contributing to the understanding of combinatorial game theory in graph structures.

Contribution

It introduces and analyzes the impartial geodetic building games on graphs, determining nim-values for several graph families and expanding game theory in graph convexity contexts.

Findings

Nim-values are determined for specific graph families.

Strategies depend on the structure of the graph.

The games' outcomes vary with graph topology.

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.