Analysis of Odd/odd vertex removal games on special graphs
Oliver Kr\"uger

TL;DR
This paper studies the Odd/odd vertex removal game on special graphs, proving a parity-based Grundy value characterization for bipartite graphs and confirming the existence of graphs for all Grundy values.
Contribution
It generalizes a conjecture by Shelton by showing bipartite graphs have Grundy values 0 or 1 based on edge parity and confirms graphs exist for every Grundy value.
Findings
Bipartite graphs have Grundy value 0 or 1 depending on edge parity
Confirmed the existence of graphs for every Grundy value
Extended previous conjectures on vertex removal games
Abstract
We analyze the Odd/odd vertex removal game introduced by P. Ottaway. We prove that every bipartite graph has Grundy value 0 or 1 only depending on the parity of the number of edges in the graph, which is a generalization of a conjecture of K. Shelton. We also answer a question originally posed by both Shelton and Ottaway about the existance of graphs for every Grundy value. We prove that this is indeed the case.
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Taxonomy
TopicsArtificial Intelligence in Games · Advanced Graph Theory Research · Game Theory and Applications
