The Closed Geodetic Game: algorithms and strategies

TL;DR

This work analyzes the Closed Geodetic Game, an impartial game based on geodetic closures, and derives Sprague-Grundy values for key graph families while providing efficient algorithms for structured classes. It establishes general results for Cartesian products, develops symmetry-based insights for paths, cycles, and grids, and delivers linear-time and quadratic-time dynamic-programming approaches for block graphs and cacti, respectively. The findings illuminate how graph structure—via articulation points, cliques, and cycle composition—drives game outcomes and enable scalable computation of nim-values. Collectively, the paper advances both theoretical understanding and practical toolkits for analyzing Closed Geodetic Game across constrained graph classes, with potential extensions to additional graph products and broader families.

Abstract

The geodetic closure of a set S of vertices of a graph is the set of all vertices in shortest paths between pairs of vertices of S. A set S of vertices in a graph is geodetic if its geodetic closure contains all the vertices of the graph. Buckley introduced in 1984 the idea of a game where two players construct together a geodetic set by alternately selecting vertices, the game ending when all vertices are in the geodetic closure. The Geodetic Game was then studied in 1985 by Buckley and Harary, and allowed players to select vertices already in the geodetic closure of the current set. We study the more natural variant, also introduced in 1985 by Buckley and Harary and called the Closed Geodetic Game, where the players alternate adding to a set S vertices that are not in the geodetic closure of S, until no move is available. This variant was only studied ever since for trees by Araujo et al. in 2024. We provide a full characterization of the Sprague-Grundy values of graph classes such as paths and cycles, of the outcomes of the Cartesian product of several graphs in function of their individual outcomes, and give polynomial-time algorithms to determine the Sprague-Grundy values of cactus and block graphs.

Figures (8)

• Figure 1: Black vertices belong to the set $S$, crossed vertices belong to the geodetic closure $(S)$, white vertices are uncovered vertices yet. However, if we add the dotted vertices to $S$, then $V=(S)$, i.e., $S$ becomes a geodetic set.
• Figure 2: An example of a graph with different outcomes for Geodetic Game and Closed Geodetic Game.
• Figure 3: Selecting an articulation point is equivalent to splitting the graph and playing independently on the components.
• Figure 4: Selecting $u_{k-i}$ and $u_{k+1+i}$ are equivalent moves on $C_{2k+1},\{u_0\}$ (here, with $k=5$ and $i=1$).
• Figure 5: Selecting a vertex in a clique in a block graph allows us to decompose the graph as follows (here, each $B_i^x$ is a clique).

Theorems & Definitions (26)

• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof