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43
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 39 (9 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.
Constructions for Cubic Graphs With Large Girth
 Electronic Journal of Combinatorics
, 1998
"... The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a ..."
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Cited by 35 (0 self)
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The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a graph whose girth is exactly g. The values of ¯ 0 (g) when 3 g 8 have been known for over thirty years. For these values of g each minimal graph is unique and, apart from the case g = 7, a simple lower bound is attained. This paper is mainly concerned with what happens when g 9, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if g = 12. A number of techniques are described, with emphasis on the construction of families of graphs fG i g for which the number of vertices n i and the girth g i are such that n i 2 cg i for some finite constant c. The optimum value of c is known to lie between 0:5 and 0:75. At the end of the p...
Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (15 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
Finding Triconnected Components By Local Replacement
, 1993
"... . We present a parallel algorithm for finding triconnected components on a CRCW PRAM. The time complexity of our algorithm is O(log n) and the processortime product is O((m + n) log log n) where n is the number of vertices, and m is the number of edges of the input graph. Our algorithm, like other ..."
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Cited by 29 (6 self)
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. We present a parallel algorithm for finding triconnected components on a CRCW PRAM. The time complexity of our algorithm is O(log n) and the processortime product is O((m + n) log log n) where n is the number of vertices, and m is the number of edges of the input graph. Our algorithm, like other parallel algorithms for this problem, is based on open ear decomposition but it employs a new technique, local replacement, to improve the complexity. Only the need to use the subroutines for connected components and integer sorting, for which no optimal parallel algorithm that runs in O(log n) time is known, prevents our algorithm from achieving optimality. 1. Introduction. A connected graph G = (V; E) is kvertex connected if it has at least (k + 1) vertices and removal of any (k \Gamma 1) vertices leaves the graph connected. Designing efficient algorithms for determining the connectivity of graphs has been a subject of great interest in the last two decades. Applications of graph connect...
Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity
 Synthesis of Parallel Algorithms
, 1992
"... This report deals with a parallel algorithmic technique that has proved to be very useful in the design of efficient parallel algorithms for several problems on undirected graphs. We describe this method for searching undirected graphs, called "open ear decomposition", and we relate thi ..."
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Cited by 25 (9 self)
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This report deals with a parallel algorithmic technique that has proved to be very useful in the design of efficient parallel algorithms for several problems on undirected graphs. We describe this method for searching undirected graphs, called "open ear decomposition", and we relate this decomposition to graph biconnectivity. We present an efficient parallel algorithm for finding this decomposition and we relate it to a sequential algorithm based on depthfirst search. We then apply open ear decomposition to obtain an efficient parallel algorithm for testing graph triconnectivity and for finding the triconnnected components of a graph.
A new graph triconnectivity algorithm and its parallelization
 Combinatorica
, 1987
"... We present a new algorithm for finding the triconnected components of an undirected graph. The algorithm is based on a method of searching graphs called ‘open ear decomposition’. A parallel implementation of the algorithm on a CRCW PRAM runs in O(log 2 n) parallel time using O(n + m) processors, whe ..."
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Cited by 25 (3 self)
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We present a new algorithm for finding the triconnected components of an undirected graph. The algorithm is based on a method of searching graphs called ‘open ear decomposition’. A parallel implementation of the algorithm on a CRCW PRAM runs in O(log 2 n) parallel time using O(n + m) processors, where n is the number of vertices and m is the number of edges in the graph.
The number of planar graphs and properties of random planar graphs
 J. Amer. Math. Soc
"... We show an asymptotic estimate for the number of labelled planar graphs on n vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs. ..."
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Cited by 19 (7 self)
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We show an asymptotic estimate for the number of labelled planar graphs on n vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs.
A Linear Time Algorithm for Triconnectivity Augmentation (Extended Abstract)
 IN PROC. 32TH ANNUAL IEEE SYMP. ON FOUNDATIONS OF COMP. SCI
, 1991
"... We consider the problem of finding a smallest set of edges whose addition triconnects an undirected graph. This is a fundamental graphtheoretic problem that has applications in designing reliable networks and faulttolerant computing. We present a linear time sequential algorithm for the problem. Th ..."
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Cited by 16 (5 self)
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We consider the problem of finding a smallest set of edges whose addition triconnects an undirected graph. This is a fundamental graphtheoretic problem that has applications in designing reliable networks and faulttolerant computing. We present a linear time sequential algorithm for the problem. This is a substantial improvement over the best previous algorithm for this problem, which runs in O(n(n+m)²) time on a graph with n vertices and m edges.
The structure of locally finite twoconnected graphs
 Electron J. Combin
, 1995
"... We expand on Tutte's theory of 3blocks for 2connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3block tree of a 2connected graph. ..."
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Cited by 8 (3 self)
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We expand on Tutte's theory of 3blocks for 2connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3block tree of a 2connected graph.
Limit Points for Average Genus (II) : 2Connected Nonsimplicial Graphs
 J. Combinatorial Theory Ser. B
, 1992
"... In part (I) of this paper, it is proved that there are no limit points for the set of values of average genus of 2connected simplicial graphs and of 3connected graphs. The need for such restrictions is now demonstrated by showing that infinitely many real numbers are limit points of values of aver ..."
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Cited by 7 (3 self)
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In part (I) of this paper, it is proved that there are no limit points for the set of values of average genus of 2connected simplicial graphs and of 3connected graphs. The need for such restrictions is now demonstrated by showing that infinitely many real numbers are limit points of values of average genus for 2connected nonsimplicial graphs. A systematic method for constructing such limit points is presented, and it is proved that this method is essentially the only way to construct limit points of values of average genus for "homeomorphically nested" 2connected graphs. April 27, 1992 1 Supported by Engineering Excellence Award from Texas A&M University. 2 Supported by ONR Contract N00014850768. CUCS02192 1 Introduction In part (I) of this paper [ChGr 1990c], we have examined the values of average genus for 2connected simplicial graphs and for 3connected graphs. We proved that in each finite real interval, there are only finitely many real numbers that are values of...