Fast Enumeration Algorithms for Non-crossing Geometric Graphs

• Published in: Discrete & computational geometry Vol. 42; no. 3; pp. 443 - 468
• Main Authors: Katoh, Naoki, Tanigawa, Shin-Ichi
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.10.2009; Springer Nature B.V
• Subjects: Algorithms; Combinatorics; Computational Mathematics and Numerical Analysis; Geometry; Graphs; Mathematics; Mathematics and Statistics; Theorems; Non-crossing geometric graphs; Enumeration; Triangulations
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-009-9164-4

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.