Analysis of Odd/odd vertex removal games on special graphs

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.