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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
2
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Two-player independent set game
The game you describe is equivalent to a generalization of Kayles, called "Node-Kayles".
Christopher King's comment, suggesting the use of nimbers, aligns with what I see in the literature on this pro …