[Paper Review] Nim on the Complete Graph

Lindsay Erickson|arXiv (Cornell University)|Oct 7, 2010

Artificial Intelligence in Games3 references3 citations

TL;DR

This paper solves Nim on the complete graph with unit edge weights by introducing the SSB (Symmetric Strategy for Bipartite) strategy, proving that the first player wins whenever there are two mutually adjacent vertices and the game starts at one of them. The key contribution is establishing that in complete graphs $K_n$ with all edge weights set to 1, the first player always has a winning strategy due to the existence of $k$-mutually adjacent vertices and the SSB strategy's effectiveness on such structures.

ABSTRACT

The game of Nim as played on graphs was introduced in Nim on Graphs I and extended in Nim on Graphs II by Masahiko Fukuyama. His papers detail the calculation of Grundy numbers for graphs under specific circumstances. We extend these results and introduce the strategy for even cycles. This paper examines a more general class of graphs by restricting the edge weight to one. We provide structural conditions for which there exist a winning strategy. This yields the solution for the complete graph.

Motivation & Objective

• To extend existing results on Nim on graphs by solving the game on complete graphs with unit edge weights.
• To identify structural conditions under which the first player has a winning strategy in Nim on graphs.
• To formalize and prove the effectiveness of the SSB (Symmetric Strategy for Bipartite) strategy in forcing a win on symmetric graph substructures.
• To generalize Fukuyama’s earlier results on Grundy numbers and edge-weighted Nim graphs to a broader class of graphs, including complete graphs.

Proposed method

• Introduce the concept of $k$-mutually adjacent vertices: two adjacent vertices with $k$ common neighbors, forming a $K_{2,k}$ subgraph.
• Define the SSB (Symmetric Strategy for Bipartite) strategy: the first player alternates moves between two mutually adjacent vertices to force the second player into a losing position.
• Use induction on $k$-mutual adjacency to prove that the first player wins on any graph containing at least two $k$-mutually adjacent vertices with $\Delta$ at one of them.
• Apply the SSB strategy to complete graphs $K_n$ with all edge weights equal to 1, showing that any two vertices in $K_n$ are $(n-2)$-mutually adjacent.
• Leverage the property that in a $p$-position, a player can always move to a $0$-position, ensuring control over game flow.
• Use graph isomorphism and option equivalence to analyze symmetric game states and reduce complex graphs to manageable substructures.

Experimental results

Research questions

• RQ1 Under what structural conditions on a graph with unit edge weights does the first player have a winning strategy in Nim?
• RQ2 Can the SSB strategy be generalized to prove first-player wins on complete graphs with unit edge weights?
• RQ3 How does $k$-mutual adjacency of vertices influence the outcome of Nim on graphs?
• RQ4 What is the role of symmetric subgraphs, such as $K_{2,k}$, in determining Grundy numbers and game outcomes?
• RQ5 Does the SSB strategy remain effective under arbitrary edge weight assignments, or is it limited to unit weights?

Key findings

• The first player has a winning strategy on any graph containing at least two $k$-mutually adjacent vertices when the game starts at one of them.
• The SSB strategy ensures that the first player can always force the second player into a symmetric, losing position by alternating moves between two mutually adjacent vertices.
• In complete graphs $K_n$ with all edge weights equal to 1, every pair of vertices is $(n-2)$-mutually adjacent, guaranteeing a first-player win for all $n > 1$.
• The result extends Fukuyama’s work on Grundy numbers by providing a structural, rather than computational, solution for a broad class of graphs.
• The SSB strategy is ineffective for arbitrary edge weight assignments, but remains valid and sufficient for unit-weight complete graphs.
• For $n \leq 7$, the first player can win $K_n$ even with arbitrary edge weights, though the SSB strategy does not apply in this case.