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{\bf Nathan Reading}
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{\bf Nim-Regularity of Graphs}
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Ehrenborg and Steingr\'\i msson defined simplicial Nim, 
and defined Nim-regular complexes to be simplicial complexes 
for which simplicial Nim has a particular type of winning 
strategy.  We completely characterize the Nim-regular graphs 
by the exclusion of two vertex-induced subgraphs, the graph 
on three vertices with one edge and the graph on five vertices 
which is complete except for one missing edge.  We show that 
all Nim-regular graphs have as their basis the set of disjoint 
unions of circuits (minimal non-faces) of the graph.


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