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Consider this game on simple graphs described by Allen Knutson: Begin by assigning a $1$ to a single node and a $0$ to each other node in the graph. Then, while such a node exists, choose a node with a value strictly less than half the sum of all nodes adjacent to it, and increment that node's value by one. We'll say this game terminates if eventually no such node exists, regardless of initial choice of assignment of $1$ and choices of node to increment at each iteration.

This game terminates for the ADE Dynkin graphs (since it's just a variation of Kostant's game that terminates for exactly the ADE Dynkin graphs), but it also terminates for other graphs. For example it terminates for any cycle graph, but not for a cycle graph plus an additional degree-$1$ node. It also doesn't terminate for the complete graph on four vertices. For which graphs does this game terminate? Does this terminate exactly for (subgraphs of) the affine ADE Dynkin diagrams like one of the comments on that answer suggests?

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The answer is yes, this game terminates for precisely the ADE Dynkin diagrams and the affine ADE Dynkin diagrams. Note that this game has been idiosyncratically named The Sponsor Game by Alexander Postnikov, as can be found in the lecture notes for his Topics in Combinatorics class.

Looking at the note The Spectral Characterization of Simply-Laced Dynkin Diagrams by Timothy Ngotiaoco also hosted on that course page, this can be proven by noting that a graph for which the Sponsor Games terminates must have the property that the eigenvalues of its adjacency matrix are at most $2$, and this is also a way of characterizing the usual and affine ADE Dynkin graphs.

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