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Asymptotic enumeration and limit laws of planar graphs
"... Abstract. We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate gn ∼ g ·n −7/2 γ n n! for the number gn of labelled planar graphs on n vertices, where γ and g are explicit computable constants. We show that the number of edges in random planar graphs ..."
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Cited by 51 (10 self)
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Abstract. We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate gn ∼ g ·n −7/2 γ n n! for the number gn of labelled planar graphs on n vertices, where γ and g are explicit computable constants. We show that the number of edges in random planar graphs is asymptotically normal with linear mean and variance and, as a consequence, the number of edges is sharply concentrated around its expected value. Moreover we prove an estimate g(q) · n −4 γ(q) n n! for the number of planar graphs with n vertices and ⌊qn ⌋ edges, where γ(q) is an analytic function of q. We also show that the number of connected components in a random planar graph is distributed asymptotically as a shifted Poisson law 1+P(ν), where ν is an explicit constant. Additional Gaussian and Poisson limit laws for random planar graphs are derived. The proofs are based on singularity analysis of generating functions and on perturbation of singularities.
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
, 1998
"... . We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than th ..."
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Cited by 45 (11 self)
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. We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 Introduction This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different tradeoffs have been proposed. The adjacencylist encoding of a graph is widely useful but requires 2mdlog ne bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted nnode tree into a string of n pairs of balanced parentheses. Since the total number of such trees is...
A Fast General Methodology For InformationTheoretically Optimal Encodings Of Graphs
, 1999
"... . We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. Specifically, a graph with property is called a graph. If satisfies certain properties, then an nnode medge graph G can be encoded by a binary string X such that (1) G and X can be obtai ..."
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Cited by 28 (3 self)
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. We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. Specifically, a graph with property is called a graph. If satisfies certain properties, then an nnode medge graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most fi(n)+o(fi(n)) bits for any continuous superadditive function fi(n) so that there are at most 2 fi(n)+o(fi(n)) distinct nnode graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from f1; : : : ; ` 1 g for ` 1 n; (5) the edges of G are labeled with labels from f1; : : : ; ` 2 ...
Lineartime succinct encodings of planar graphs via canonical orderings
 SIAM Journal on Discrete Mathematics
, 1999
"... Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, rough ..."
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Cited by 22 (6 self)
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Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5 + 2 log 3) min{n, f} −7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.
Enumeration and random realization of triangulated surfaces.arXiv:math.CO
"... We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1 ..."
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Cited by 16 (9 self)
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We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1
Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees
 In: 8 th Annual International Computing & Combinatorics Conference (COCOON). Volume 2387 of LNCS
, 2002
"... We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G. ..."
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Cited by 11 (3 self)
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We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G.
On the number of planar orientations with prescribed degrees
, 2008
"... We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α: V → N unifies many diffe ..."
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Cited by 9 (3 self)
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We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α: V → N unifies many different combinatorial structures, including the afore mentioned. We call these orientations αorientations. The main focus of this paper are bounds for the maximum number of αorientations that a planar map with n vertices can have, for different instances of α. We give examples of triangulations with 2.37 n Schnyder woods, 3connected planar maps with 3.209 n Schnyder woods and inner triangulations with 2.91 n bipolar orientations. These lower bounds are accompanied by upper bounds of 3.56 n, 8 n and 3.97 n respectively. We also show that for any planar map M and any α the number of αorientations is bounded from above by 3.73 n and describe a family of maps which have at least 2.598 n αorientations.
Counting planar graphs and related families of graphs
 In Surveys in combinatorics 2009, 169–210
, 2009
"... In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree dist ..."
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Cited by 8 (3 self)
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In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree distribution or the size of the largest kconnected component. Most of the results we present use generating functions and analytic tools.
A direct decomposition of 3connected planar graphs
 In Proceedings of the 17th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC05
, 2005
"... ABSTRACT. We present a decomposition strategy for cnets, i. e., rooted 3connected planar maps. The decomposition yields an algebraic equation for the number of cnets with a given number of vertices and a given size of the outer face. The decomposition also leads to a deterministic and polynomial ..."
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Cited by 6 (5 self)
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ABSTRACT. We present a decomposition strategy for cnets, i. e., rooted 3connected planar maps. The decomposition yields an algebraic equation for the number of cnets 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 cnets uniformly at random. Using rejection sampling, we can also sample isomorphism types of convex polyhedra, i.e., 3connected planar graphs, uniformly at random. RÉSUMÉ. Nous proposons une stratégie de décomposition pour les cartes pointées 3connexes (créseaux). Cette décomposition permet d’obtenir une équation algébrique pour le nombre de créseaux suivant le nombre de sommets et la taille de la face extèrieure. On en déduit un algorithme de complexité en temps polynomiale pour le tirage aléatoire uniforme des créseaux. En utilisant une méthode à rejet, nous obtenons aussi un algorithme de tirage aléatoire uniforme pour les graphes planaires 3connexes. 1.