Intersection graphs of convex polygons inscribed to a circle, so called polygon-circle graphs, generalize several well studied classes of graphs, e.g., interval graphs, circle graphs, circular-arc graphs and chordal graphs. We consider the question how complicated need to be the polygons in a polygon-circle representation of a graph. Let cmp(n) denote the minimum fc such that every polygon-circle graph on n vertices is the intersection graph of k-gons inscribed to the circle. We prove that cmp(n) = n - log_2 n + o(log_2 n) by showing that for every positive constant c < 1, cmp(n) ≤ n - c log n for every sufficiently large n, and by providing an explicit construction of polygon-circle graphs on n vertices which are not representable by polygons with less than n - log n - 2 log log n corners. We also show that recognizing intersection graphs of k-gons inscribed in a circle is an NP-complete problem for every fixed k ≥ 3.
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