Impartial removing games on grid graphs

TL;DR

The paper addresses nim-value analysis for impartial removing convex-hull games on grid and lattice graphs, introducing two variants TER (achievement) and DNT (avoidance) defined via the convex hull operator $[\,\cdot\,]$. It develops a unified framework using delayed gamegraphs, option-preserving maps, and a matrix/tensor reduction via an $\alpha$-map to reduce grid games to tractable $3\times3$ matrix games, from which exact nim-values are derived: $\text{nim}(\text{DNT}(P_m\Box P_n))=\text{pty}(mn)$ and $\text{nim}(\text{TER}(P_m\Box P_n))=\text{pty}(mn)$ for grid graphs. The analysis extends to $d$-dimensional lattice graphs $\Gamma=P_{n_1}\Box\cdots\Box P_{n_d}$, yielding $\text{nim}(\text{DNT}(\Gamma))=\text{pty}(n_1\cdots n_d)$ and establishing partial results (with a conjecture) for $\text{nim}(\text{TER}(\Gamma))$; central-reflection strategies and tensor-generalizations underpin these results. Overall, the authors provide a practical, parity-driven methodology for computing nim-values in removing convex-hull games on large graph families and highlight directions for complete characterization in higher dimensions.

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.

Figures (11)

• Figure 2.1: The graphs $P_{3}\Box P_{4}$ and $P_{2}\Box P_{2}\Box P_3$.
• Figure 2.2: Convex hulls of subsets of $P_3\Box P_5$.
• Figure 2.3: Four minimal vertex subsets $S$ of $P_2\Box P_2\Box P_3$ satisfying $[S]=V$.
• Figure 2.4: Representative quotient gamegraph for $\text{DNT}(P_2\Box P_3)$. Vertices depicted by $\bullet$ are already selected, while vertices depicted by $\circ$ are still unselected.
• Figure 3.1: Sum $\mathsf{G}\Box\mathsf{D}_{3}$ and delayed $\mathsf{G}{\hbox{\larger[-.75]$\sqcap$}}\mathsf{D}_{3}$ gamegraphs.

Theorems & Definitions (38)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 3.1
• Proposition 3.2
• proof
• Corollary 3.3
• Example 3.4
• Example 4.1
• Proposition 4.2