Impartial removing games on grid graphs

TL;DR

This paper analyzes two impartial games played on grid graphs involving convex hulls, determining their nim-numbers and providing insights into game strategies using advanced graph-theoretic tools.

Contribution

It introduces a detailed analysis of convex hull-based games on grid graphs and computes their nim-numbers, extending understanding of combinatorial game theory on lattice structures.

Findings

Nim-numbers for the achievement game on grid graphs are determined.

Nim-numbers for the avoidance game on grid graphs are established.

Results extend to certain lattice graphs beyond grid graphs.

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 subset. The convex hull of a set of vertices is the smallest convex set containing the set. We study two games in which two players take turns selecting vertices of a graph until the convex hull of the remaining unselected vertices is too small. The last player to move is the winner. The achievement game ends when the convex hull of the unselected vertices does not contain every vertex in the graph. In the avoidance game, the convex hull of the remaining vertices must contain every vertex. We determine the nim-number of these games for the family of grid graphs. We also provide some results for lattice graphs. Key tools in this analysis are delayed gamegraphs, option preserving maps, and case analysis diagrams.