Two results on intersection graphs of polygons

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 k 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) less than or equal to n - clog 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 greater than or equal to 3.