On Intersection Graphs of Convex Polygons

Since an image can easily be modeled by its adjacency graph, graph theory and algorithms on graphs are widely used in image processing. Of particular interest are the problems of estimating the number of the maximal cliques in a graph and designing algorithms for their computation, since these are found relevant to various applications in image processing and computer graphics. In the present paper we study the maximal clique problem on intersection graphs of convex polygons, which are also applicable to imaging sciences. We present results which refine or improve some of the results recently proposed in [18]. Thus, it was shown therein that an intersection graph of n convex polygons whose sides are parallel to k different directions has no more than n(2k) maximal cliques. Here we prove that the number of maximal cliques does not exceed n k. Moreover, we show that this bound is tight for any fixed k. Algorithmic aspects are discussed as well.