Chomp on Graphs and Subsets

The game subset take-away begins with a simplicial complex 4. Two players take turns removing any element of 4 as 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.