Vertex Nim played on graphs

TL;DR

This paper studies the computational complexity of Vertex Nim played on graphs, providing efficient algorithms for undirected graphs and certain directed graphs, and improving previous results on related games.

Contribution

It introduces polynomial-time algorithms for determining winning positions in Vertex Nim on undirected graphs and specific directed graphs, advancing prior exponential-time methods.

Findings

Undirected Vertex Nim can be decided in quadratic time.

The algorithm applies to Vertex NimG, improving Stockman's exponential algorithm.

Polynomial-time solutions are found for certain directed graphs like circuits and self-loop digraphs.

Abstract

Given a graph G with positive integer weights on the vertices, and a token placed on some current vertex u, two players alternately remove a positive integer weight from u and then move the token to a new current vertex adjacent to u. When the weight of a vertex is set to 0, it is removed and its neighborhood becomes a clique. The player making the last move wins. This adaptation of Nim on graphs is called Vertexnim, and slightly differs from the game Vertex NimG introduced by Stockman in 2004. Vertexnim can be played on both directed or undirected graphs. In this paper, we study the complexity of deciding whether a given game position of Vertexnim is winning for the first or second player. In particular, we show that for undirected graphs, this problem can be solved in quadratic time. Our algorithm is also available for the game Vertex NimG, thus improving Stockman's exptime algorithm. In the directed case, we are able to compute the winning strategy in polynomial time for several instances, including circuits or digraphs with self loops.