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.

The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined 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...

Pointed pseudo-triangulations 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.

. 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 processor-time 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 k-vertex 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...

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.

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 depth-first 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.

We consider the problem of finding a smallest set of edges whose addition triconnects an undirected graph. This is a fundamental graph-theoretic 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.

In part (I) of this paper, it is proved that there are no limit points for the set of values of average genus of 2-connected simplicial graphs and of 3-connected 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 2-connected non-simplicial 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" 2-connected graphs. April 27, 1992 1 Supported by Engineering Excellence Award from Texas A&M University. 2 Supported by ONR Contract N00014-85-0768. CUCS-021-92 1 Introduction In part (I) of this paper [ChGr 1990c], we have examined the values of average genus for 2-connected simplicial graphs and for 3-connected graphs. We proved that in each finite real interval, there are only finitely many real numbers that are values of...

We expand on Tutte's theory of 3-blocks for 2-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3-block tree of a 2-connected graph.

A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are given an embedding of K in some surface. The embedding extension problem asks whether it is embedding extension problem possible to extend the embedding of K to an embedding of G in the same surface, and any such embedding is an embedding extension of K to G. An embedding extension obstruction for embedding extensions is a subgraph\Omega of G \Gamma E(K) such that obstruction the embedding of K cannot be extended to K [ \Omega\Gamma The obstruction is small small if K [\Omega is homeomorphic to a graph with a small number of edges. If\Omega is small, then it is easy to verify (for example, by checking all the possibilities Supported in part by the Ministry of Science and Technolo...