Fast Enumeration Algorithms for Non-crossing Geometric Graphs

A non-crossing geometric graph is a graph embedded on a set of points in the plane with non-crossing straight line segments. In this paper we present a general framework for enumerating non-crossing geometric graphs on a given point set. Applying our idea to specific enumeration problems, we obtain faster algorithms for enumerating plane straight-line graphs, non-crossing spanning connected graphs, non-crossing spanning trees, and non-crossing minimally rigid graphs. Our idea also produces efficient enumeration algorithms for other graph classes, for which no algorithm has been reported so far, such as non-crossing matchings, non-crossing red-and-blue matchings, non-crossing k-vertex or k-edge connected graphs, or non-crossing directed spanning trees. The proposed idea is relatively simple and potentially applies to various other problems of non-crossing geometric graphs.