Maximal Cliques in Unit Disk Graphs: Polynomial Approximation

We consider the problem of generating all maximal cliques in an unit disk graph. General algorithms to find all maximal cliques are exponential, so we rely on a polynomial approximation. Our algorithm makes use of certain key geographic structures of these graphs. For each edge, we limit the set of vertices that may form cliques with this as the longest edge. We then consider several characteristic shapes determined by that edge, and prove that all cliques having this as the longest edge, are included in one of the sets of vertices contained in these shapes. Our algorithm works in O(m¢ 2 ) time and generates O(m¢) cliques, where m is the number of edges in the graph and ¢ is its maximum degree. We also provide a modified version of the algorithm which improves the performance in many cases, albeit without affecting the worst case running time.