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In this paper, we give the sharp upper bound for the number of vertices withpositive curvature in a planar graph with nonnegative combinatorial curvature.Based on this, we show that the automorphism group of a planar---possiblyinfinite---graph with nonnegative combinatorial curvature and positive totalcurvature is a finite group, and give an upper bound estimate for the order ofthe group.

The set of vertices with positive curvature in a planar graph with nonnegative curvature