Determining the maximum clique size in large random geometric graphs

Abstract

For a geometric graph on a topology, we employ an iterated version of the smallest last coloring algorithm to obtain either the largest clique or very tight bounds on its size. The procedure is topology independent and runs in two phases, global and local. We exploit pruning methods in building the geometric graphs and in limiting the number of sub-graphs searched. We have performed experiments using random geometric graphs in the plane and on the sphere, and provide computational results. The time complexity of the algorithm is O(|E| ×|V| + |V| <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) where |E| and |V| are the cardinalities of a geometric graph's edge set and vertex set, respectively.