On the Number of Plane Geometric Graphs

We investigate the number of plane geometric, i.e., straight-line, graphs, a set S of n points in the plane admits. We show that the number of plane geometric graphs and connected plane geometric graphs as well as the number of cycle-free plane geometric graphs is minimized when S is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide a unified proof that the cardinality of any family of acyclic graphs (for example spanning trees, forests, perfect matchings, spanning paths, and more) is minimized for point sets in convex position.In addition we construct a new maximizing configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ*$${{(\sqrt{72}\,}^n)}$$ = Θ*(8.4853n) triangulations (omitting polynomial factors), improving the previously known best maximizing examples.

[1]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[2]  Oswin Aichholzer,et al.  The point set order type data base: A collection of applications and results , 2001, CCCG.

[3]  Raimund Seidel,et al.  A better upper bound on the number of triangulations of a planar point set , 2003, J. Comb. Theory, Ser. A.

[4]  Ferran Hurtado,et al.  On the number of plane graphs , 2006, SODA '06.

[5]  O. Aichholzer,et al.  On the Crossing Number of Complete Graphs , 2005, Computing.

[6]  Adrian Dumitrescu,et al.  On two lower bound constructions , 1999, CCCG.

[7]  E. Szemerédi,et al.  Crossing-Free Subgraphs , 1982 .

[8]  Oswin Aichholzer,et al.  Abstract order type extension and new results on the rectilinear crossing number , 2005, EuroCG.

[9]  Günter Rote,et al.  Counting triangulations and pseudo-triangulations of wheels , 2001, CCCG.

[10]  Marc Noy,et al.  Lower bounds on the number of crossing-free subgraphs of KN , 2000, Comput. Geom..

[11]  Franz Aurenhammer,et al.  Enumerating Order Types for Small Point Sets with Applications , 2002, Order.

[12]  Franz Aurenhammer,et al.  Convexity minimizes pseudo-triangulations , 2002, CCCG.

[13]  Marc Noy,et al.  A lower bound on the number of triangulations of planar point sets , 2004, Comput. Geom..

[14]  Bettina Speckmann,et al.  On the Number of Pseudo-Triangulations of Certain Point Sets , 2003, CCCG.

[15]  F. Thomas Leighton,et al.  Complexity Issues in VLSI , 1983 .

[16]  Francisco Santos,et al.  The polytope of non-crossing graphs on a planar point set , 2004, ISSAC '04.

[17]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[18]  Micha Sharir,et al.  Random triangulations of planar point sets , 2006, SCG '06.

[19]  Micha Sharir,et al.  On the number of crossing-free matchings, (cycles, and partitions) , 2006, SODA '06.

[20]  Philippe Flajolet,et al.  Analytic combinatorics of non-crossing configurations , 1999, Discret. Math..