[Paper Review] Chomp on Graphs and Subsets

Tirasan Khandhawit, Lynnelle Ye|arXiv (Cornell University)|Jan 14, 2011

Artificial Intelligence in Games5 references3 citations

TL;DR

This paper introduces a generalized symmetry-based reduction method for analyzing subset take-away and graph chomp games on simplicial complexes and graphs. It provides a complete classification of nim-values for complete $n$-partite graphs and all bipartite graphs, and offers partial results for odd-cycle pseudotrees, significantly extending prior work on special cases like complete graphs and forests.

ABSTRACT

The game subset take-away begins with a simplicial complex Δ. Two players take turns removing any element of Δas well as all other elements which contain it, and the last player able to move wins. Graph Chomp is a special case of subset take-away played on a simplicial complex with only vertices and edges. The game has previously only been analyzed for complete graphs, forest graphs, and very small special cases of higher-dimensional simplicial complexes. We generalize a common method of reducing some game positions to simpler ones by symmetry and provide a complete analysis of complete n-partite graphs for arbitrary n and all bipartite graphs. Finally, we give partial results for odd-cycle pseudotrees, which are non-bipartite graphs with a single cycle.

Motivation & Objective

• To extend the analysis of subset take-away and graph chomp beyond known special cases such as complete graphs and forests.
• To develop a general reduction technique based on symmetry that simplifies game positions in poset games.
• To fully classify the nim-values of complete $n$-partite graphs and all bipartite graphs.
• To provide partial characterization of nim-values for non-bipartite pseudotrees with odd cycles.
• To lay foundational results for understanding more complex poset games, including chocolate bar chomp.

Proposed method

• Introduces a new symmetry-based reduction method applicable to subset take-away games on simplicial complexes, generalizing prior approaches.
• Applies the reduction method to complete $n$-partite graphs, proving they are first-player wins for $n \geq 3$ and determining their nim-values.
• Develops a theorem to compute the nim-value of all bipartite graphs using structural decomposition and inductive reasoning.
• Uses induction on subgraphs to classify nim-values of pseudotrees with odd cycles, identifying base cases and transition rules.
• Employs nim-value analysis via disjunctive game theory, computing $g(G) = \text{mex}\{g(G')\}$ for all options $G'$ of $G$.
• Analyzes special configurations such as hairballs (cycles with tails), cycles joined by vertices, and cycles with additional paths between nonadjacent vertices.

Experimental results

Research questions

• RQ1 Can a general symmetry-based reduction method be developed to simplify subset take-away and graph chomp game positions beyond known special cases?
• RQ2 What are the nim-values of complete $n$-partite graphs for arbitrary $n$?
• RQ3 How can the nim-values of all bipartite graphs be systematically determined?
• RQ4 What is the structure of nim-values for pseudotrees with odd cycles, particularly those with multiple tails or additional paths?
• RQ5 Under what conditions can a graph be reduced to a known game position via symmetry or structural decomposition?

Key findings

• Complete $n$-partite graphs for $n \geq 3$ are first-player wins, and their nim-values are fully characterized using the proposed reduction method.
• All bipartite graphs have their nim-values completely determined by a new structural theorem, enabling full classification.
• For pseudotrees with odd cycles, the paper identifies three base case configurations where the nim-value is 4, and proves that such graphs have nim-value 4 if they are not in these forms.
• Corollary 2 shows that a graph with $V$ vertices and $V+1$ edges, where one vertex has degree 4 and others have degree 2, has nim-value 1.
• Corollary 3 establishes that a cycle with two nonadjacent vertices connected by an additional path has nim-value 1 if the total number of vertices is odd, and 2 if even.
• The paper proves that certain configurations, such as a cycle with two tails of consecutive lengths on the same vertex, can be reduced to nim-value 0 via deletion of the second-longest tail’s leaf, enabling inductive analysis.