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Random planar graphs
 JOURNAL OF COMBINATORIAL THEORY, SERIES B 93 (2005) 187 –205
, 2005
"... We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive i ..."
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Cited by 46 (9 self)
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We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability Rn has linearly many vertices of a given degree, in each embedding Rn has linearly many faces of a given size, and Rn has exponentially many automorphisms.
Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
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Cited by 39 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
Generating Labeled Planar Graphs Uniformly at Random
, 2003
"... We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3con ..."
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Cited by 25 (7 self)
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We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.
Planar graphs, via wellorderly maps and trees
 In 30 th International Workshop, Graph  Theoretic Concepts in Computer Science (WG), volume 3353 of Lecture Notes in Computer Science
, 2004
"... Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequ ..."
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Cited by 20 (4 self)
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Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worstcase, a rate of 4.91 bits per node and of 2.82 bits per edge. 1
Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation
, 2003
"... In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n node ..."
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Cited by 12 (1 self)
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In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive: an enumeration formula, and an asymptotic of 2 3n (log n) ; an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worstcase constant time and neighbors query of a ddegree node in worstcase O(d) time...
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 12 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
Transversal structures on triangulations, with application to straight line drawing
 LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We define and study a structure called transversal edgepartition related to triangulations without non empty triangles, which is equivalent to the regular edge labeling discovered by Kant and He. We study other properties of this structure and show that it gives rise to a new straightline drawing ..."
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Cited by 12 (5 self)
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We define and study a structure called transversal edgepartition related to triangulations without non empty triangles, which is equivalent to the regular edge labeling discovered by Kant and He. We study other properties of this structure and show that it gives rise to a new straightline drawing algorithm for triangulations without non empty triangles, and more generally for 4connected plane graphs with at least 4 border vertices. Taking uniformly at random such a triangulation with 4 border vertices and n vertices, the size of the grid is almost surely n
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
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Cited by 6 (1 self)
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Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �
Sampling unlabeled biconnected planar graphs
 in the Proceedings of the 16th Annual International Symposium on Algorithms and Computation (ISAAC’05), 2005, Springer LNCS 3827, 593 – 603
"... Abstract. We present an expected polynomial time algorithm to generate a 2connected unlabeled planar graph uniformly at random. To do this we first derive recurrence formulas to count the exact number of rooted 2connected planar graphs, based on a decomposition along the connectivity structure. Fo ..."
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Cited by 4 (4 self)
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Abstract. We present an expected polynomial time algorithm to generate a 2connected unlabeled planar graph uniformly at random. To do this we first derive recurrence formulas to count the exact number of rooted 2connected planar graphs, based on a decomposition along the connectivity structure. For 3connected planar graphs we use the fact that they have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sensereversing or a poleexchanging automorphism. We prove a bijection between such symmetric objects and certain colored networks. These colored networks can again be decomposed along their connectivity structure. All the numbers can be evaluated in polynomial time by dynamic programming. To generate 2connected unlabeled planar graphs without a root uniformly at random we apply rejection sampling and obtain an expected polynomial time algorithm. 1