A direct decomposition of 3-connected planar graphs

We present a decomposition strategy for c-nets, i. e., root ed 3-connected planar maps. The decomposition yields an algebraic equation for the number of c-nets with a given number of vertices and a given size of the outer face. The decomposition also leads to a deterministic and polynomial time algorithm to sample c-nets uniformly at random. Using rejection sampling, we can also sample isomorphism types of convex polyhedra, i.e., 3-connected planar graphs, uniformly at r andom. R ´ ESUM´ E. Nous proposons une strategie de decomposition pour les cartes pointees 3- connexes (c-reseaux). Cette decomposition permet d'obtenir une ´ eq uation algebrique pour le nombre de c-reseaux suivant le nombre de sommets et la taille de la face exterieure. On en deduit un algorithme de complexite en temps polynomiale pour le tirage aleatoire uniforme des c-reseaux. En utilisant une methoderejet, nous obtenons aussi un algo- rithme de tirage aleatoire uniforme pour les graphes planaires 3-connexes.

[1]  H. Hahn.  Encyklopädie der mathematischen Wissenschaften , 1928 .

[2]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[3]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[4]  R. Mullin,et al.  The enumeration of c-nets via quadrangulations , 1968 .

[5]  Robert E. Tarjan,et al.  A V² Algorithm for Determining Isomorphism of Planar Graphs , 1971, Inf. Process. Lett..

[6]  P. J. Federico.  THE NUMBER OF POLYHEDRA , 1975 .

[7]  Nicholas C. Wormald,et al.  On the Tumber of Planar Maps , 1981, Canadian Journal of Mathematics.

[8]  Timothy R. S. Walsh,et al.  Counting non-isomorphic three-connected planar maps , 1982, J. Comb. Theory, Ser. B.

[9]  T. C. Hu,et al.  Combinatorial algorithms , 1982 .

[10]  Edward A. Bender,et al.  The asymptotic enumeration of rooted convex polyhedra , 1984, J. Comb. Theory, Ser. B.

[11]  Edward A. Bender,et al.  Almost all Convex Polyhedra are Asymmetric , 1985, Canadian Journal of Mathematics.

[12]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[13]  Philippe Flajolet,et al.  A Calculus for the Random Generation of Labelled Combinatorial Structures , 1994, Theor. Comput. Sci..

[14]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[15]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[16]  Gilles Schaeffer.  Conjugaison d'arbres et cartes combinatoires aléatoires , 1998 .

[17]  Gilles Schaeffer,et al.  Random sampling of large planar maps and convex polyhedra , 1999, STOC '99.

[18]  Alain Denise,et al.  Uniform Random Generation of Decomposable Structures Using Floating-Point Arithmetic , 1999, Theor. Comput. Sci..

[19]  Kyriakos Kalorkoti.  ALGEBRAIC COMPLEXITY THEORY (Grundlehren der Mathematischen Wissenschaften 315) , 1999 .

[20]  Philippe Flajolet,et al.  Random maps, coalescing saddles, singularity analysis, and Airy phenomena , 2001, Random Struct. Algorithms.

[21]  Philippe Flajolet,et al.  ANALYTIC COMBINATORICS — SYMBOLIC COMBINATORICS , 2002 .

[22]  Manuel Bodirsky,et al.  Generating Labeled Planar Graphs Uniformly at Random , 2003, ICALP.

[23]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

[24]  Éric Fusy,et al.  Dissections and trees, with applications to optimal mesh encoding and to random sampling , 2005, SODA '05.

[25]  Manuel Bodirsky,et al.  Sampling Unlabeled Biconnected Planar Graphs , 2005, ISAAC.

[26]  D. Johannsen,et al.  Sampling Rooted 3-Connected Planar Graphs in Deterministic Polynomial Time , 2006 .