3-connected Components Research Articles

Excluded minors for the Klein bottle II. Cascades

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus k or for embeddings into nonorientable surface of genus k are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained.

Decomposing the percolation backbone reveals novel scaling laws of the current distribution

The distribution of currents on critical percolation clusters is the fundamental quantity describing the transport properties of weakly connected systems. Nevertheless, its finite-size extrapolation is still one of the outstanding open questions concerning disordered media. By hierarchically decomposing the 3-connected components of the backbone, we disclose that the current distribution is determined from two distributions, namely, the one corresponding to the number of bonds in each level and another one corresponding to the factors by which the current is reduced, when going from one level to the next. The first distribution follows a finite-size scaling, while the second is a power law with an exponent consistent with 3/4 in two dimensions. The standard hierarchical model for the backbone is too simple to reproduce this complex scenario. Our new decomposition method of the backbone also allows to calculate much smaller currents than before, attaining a precision of 10−35 and systems of size L = 81922. Moreover, our method is not restricted to electric currents on critical percolation clusters but could also be applied to other transport problems on sparse graphs including fluid flow and car traffic.

GPART: Partitioning Maximal Redundant Rigid and Maximal Global Rigid Components in Generic Distance Graphs

Partitioning the Maximal Redundant Rigid Components (MRRC) and Maximal Global Rigid Components (MGRC) in generic 2D graphs are critical problem for network structure analysis, network localizability detection, and localization algorithm design. This article presents efficient algorithms to partition MRRCs and MGRCs and develops an open-sourced toolbox, GPART, for these algorithms to be conveniently used by the society. We firstly propose conditions and an efficient algorithm to merge the over-constrained regions to form the maximal redundant rigid components (MRRC). The detected MRRCs are proved to be maximal and all the MRRCs are guaranteed to be detected. The time to merge the over-constrained regions is linear to the number of nodes in the over-constrained components. To detect MGRCs, the critical problem is to decompose 3-connected components in each MRRC. We exploit SPQR-tree based method and design a local optimization algorithm, called MGRC_acce to prune the unnecessary decomposition operations so that the SPQR-tree functions can be called much less number of times. We prove the MGRCs can be detected inside MRRCs using at most O(mn ) time. Then a GPART toolbox is developed and extensively tested in graphs of different densities. We show the proposed MRRC and MGRC detection algorithms are valid and MGRC_acce greatly outperforms the direct SPQR-tree based decomposition algorithm. GPART is outsourced at https://github.com/inlab-group/gpart .

Random cubic planar graphs converge to the Brownian sphere

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney’s theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.

Enumeration of rooted 3-connected bipartite planar maps

Abstract We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B (2011)]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2- and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form t · n −5/2 γ n , where γ = ρ −1 ≈ 2.40958 and ρ ≈ 0.41501 is an algebraic number of degree 10.

The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs

The Weisfeiler-Leman procedure is a widely-used technique for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into 2- and 3-connected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. This implies that the dimension of the algorithm needed to distinguish two given non-isomorphic graphs is at most the dimension required to distinguish non-isomorphic 3-connected components of the graphs (assuming dimension at least 2). To obtain our decomposition result, we show that, for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of the class of graphs of treewidth at most k. Using a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.

Maximum cycle packing using SPR-trees

Let G = ( V , E ) be an undirected multigraph without loops. The maximum cycle packing problem is to find a collection Z * = { C 1 , ..., C s } of edge-disjoint cycles C i subset G of maximum cardinality v ( G ) . In general, this problem is NP-hard. An approximation algorithm for computing v ( G ) for 2-connected graphs is presented, which is based on splits of G . It essentially uses the representation of the 3-connected components of G by its SPR-tree. It is proved that for generalized series-parallel multigraphs the algorithm is optimal, i.e. it determines a maximum cycle packing Z * in linear time.

Using SPQR-trees to speed up recognition algorithms based on 2-cutsets

Several well-studied classes of graphs admit structural characterizations via proper 2-cutsets which lead to polynomial-time recognition algorithms. The algorithms so far obtained for those recognition problems do not guarantee linear-time complexity. The bottleneck to those algorithms is the Ω(nm)-time complexity to fully decompose by proper 2-cutsets a graph with n vertices and m edges. In the present work, we investigate the 3-connected components of a graph and propose the use of the SPQR-tree data structure to obtain a fully decomposed graph in linear time. As a consequence, we show that the recognition of chordless graphs and of graphs that do not contain a propeller as a subgraph can be done in linear time, answering questions in the existing literature.

Computing with Tangles

Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing “$k$-connected components” of a graph (though in a twisted way). An interesting aspect of tangles is that they can be defined not only for graphs, but more generally for arbitrary connectivity functions (that is, integer-valued submodular and symmetric set functions). However, tangles are difficult to deal with algorithmically. To start with, it is unclear how to represent them, because they are families of separations and, as such, may be exponentially large. Our first contribution is a data structure for representing and accessing all tangles of a graph up to some fixed order. Using this data structure, we can prove an algorithmic version of a very general structure theorem due to Carmesin et al. (for graphs) and Hundertmark (for arbitrary connectivity functions) that yields a canonical tree decomposition whose parts correspond to the maximal tangles. This may be viewed as a generalization of the decomposition of a graph into its 3-connected components.

Using SPQR-trees to speed up algorithms based on 2-cutset decompositions

We propose the use of SPQR-trees as a data structure to encode the 3-connected components of a graph and to obtain linear-time recognition algorithms for graph classes structurally characterized by 2-cutset decompositions.

The distribution of the number of small cuts in a random planar triangulation

We enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with $n+3$ vertices is asymptotically normal with mean $(10/27)n$ and variance $(320/729)n$, and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with $n+2$ vertices is asymptotically normal with mean $(8/27)n$ and variance $(152/729)n$. We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with $n+2$ vertices is asymptotically normal with mean $n/3$ and variance $\frac{8}{ 27}n$ .

Structure and enumeration of two-connected graphs with prescribed three-connected components

We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc ( g ) associated with any 2-connected graph g, whose white vertices are the 3 -components of g (3-connected components or polygons) and black vertices are bonds arising from separating pairs of vertices of g and linking together the 3-components. Two fundamental species relationships on graphs and networks follow from this construction. The first one is a dissymmetry theorem which leads to the expression of the class B = B ( F ) of 2-connected graphs, all of whose 3-connected components belong to a given class F of 3-connected graphs, in terms of various rootings of B . The second one is a functional equation which characterizes the corresponding class R = R ( F ) of two-pole networks all of whose 3-connected components are in F . All the rootings of B are then expressed in terms of F and R . There follow corresponding identities for all the associated power series, in particular, the edge index series. These results are expressed in terms of species of structures. Using these results we enumerate several classes of labelled and unlabelled graphs, including series-parallel graphs, 2-connected planar graphs, K 3 , 3 -free 2-connected graphs and K 3 , 3 -free projective planar and toroidal graphs.

A Complete Grammar for Decomposing a Family of Graphs into 3-Connected Components

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family ${\cal G}$ of graphs (with some stability conditions) in terms of the subfamily ${\cal G}_3$ of graphs in ${\cal G}$ that are 3-connected (until now, such a general grammar was only known for the decomposition into $2$-connected components). As a byproduct, our grammar yields an explicit system of equations to express the series counting a (labelled) family of graphs in terms of the series counting the subfamily of $3$-connected graphs. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar and associated equation system, but has the considerable advantage of avoiding the difficult integration steps that appear with other approaches, in particular in recent work by Giménez and Noy on counting planar graphs. As a main application we recover in a purely combinatorial way the analytic expression found by Giménez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter. Finally, our grammar applies also to the case of unlabelled structures, since the dissymetry theorem takes symmetries into account. Even if there are still difficulties in counting unlabelled 3-connected planar graphs, we think that our grammar is a promising tool toward the asymptotic enumeration of unlabelled planar graphs, since it circumvents some difficult integral calculations.

Graph classes with given 3-connected components: asymptotic counting and critical phenomena

Fix a family T of 3-connected graphs, and let G be the class of graphs whose 3connected components are the graphs in T . We present a general framework for analyzing such graph classes based on singularity analysis of generating functions. This generalizes previously studied cases such as planar graphs and series-parallel graphs. We provide a general theorem for the asymptotic number of graphs in G, based on the singularities of the exponential generating function associated to T . For some of the classes under study we show the presence of critical phenomena as the edge density in the class varies.

Generating labeled planar graphs uniformly at random

We present a deterministic polynomial time algorithm to sample a labeled planar graph uniformly at random. Our approach uses recursive formulae for the exact number of labeled planar graphs with n vertices and m edges, based on a decomposition into 1-, 2-, and 3-connected components. We can then use known sampling algorithms and counting formulae for 3-connected planar graphs.

The monadic second-order logic of graphs XVI : Canonical graph decompositions

This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decomposition and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs having the same cycle matroid as a given 2-connected graph can be defined from this graph by Monadic Second-Order formulas.

Decomposing Infinite 2-Connected Graphs into 3-Connected Components

In the 1960's, Tutte presented a decomposition of a 2-connected finite graph into 3-connected graphs, cycles and bonds. This decomposition has been used to reduce problems on 2-connected graphs to problems on 3-connected graphs. Motivated by a problem concerning accumulation points of infinite planar graphs, we generalize Tutte's decomposition to include all infinite 2-connected graphs.

On testing consecutive-ones property in parallel

The consecutive-ones property problem has many important applications in the field of discrete algorithms, including the physical mapping problem in computational molecular biology. A (0, 1)-matrix is said to satisfy the consecutive-ones property if there is a permutation of the rows of the matrix such that in each column all non-zero entries are adjacent. The problem of determining such a permutation, if one exists, is the consecutive-ones property problem. The classic algorithm for solving this problem is a linear time sequential algorithm of Booth and Lueker (1976) which is known to be based on the PQ-tree data structure. In this paper we present a new algorithm for this problem using a divide-and-conquer method that employs a graph-theoretic data structure known as Tutte decomposition, i.e., decomposition of graphs into 3-connected components. Our algorithm enjoys the property that it efficiently parallelizes using the standard PRAM parallel computational model, while avoiding the complex implementations associated with PQ-trees. Our algorithm is more work efficient than previous parallel solutions, improving on the known processor bounds.

Parallel Tree Contraction Part 2: Further Applications

This paper applies the parallel tree contraction techniques developed in Miller and Reif’s paper [Randomness and Computation, Vol. 5, S. Micali, ed., JAI Press, 1989, pp. 47’72] to a number of fundamental graph problems. The paper presents an $O(\log n)$ time and $n / \log n$ processor, a 0-sided randomized algorithm for testing the isomorphism of trees, and an $O(\log n)$ time, n algorithm for maximal subtree isomorphism and for common subexpression elimination. An O(log n) time, n-processor algorithm for computing the canonical forms of trees and subtrees is given. An Olog n time algorithm for computing the tree of 3-connected components of a graph, an $O(\log ^2 n)$ time algorithm for computing an explicit planar embedding of a planar graph, and an $O(\log ^3 n)$ time algorithm for computing a canonical form for a planar graph are also given. All these latter algorithms use only $n^{O(1)} $ processors on a Parallel Random Access Machine (PRAM) model with concurrent writes and concurrent reads.

An Application of Duality to Edge-Deletion Problems

For a property $\pi $ on graphs, the corresponding edge-deletion problem $P_{{\text{ED}}} (\pi )$ (on planar graphs) is stated as follows: given a (planar) graph G, find a set of edges of minimum cardinality whose deletion results in a graph satisfying $\pi $. We show that the edge-deletion problem $P_{{\text{ED}}} (\pi )$ on planar graphs is NP-hard if $\pi $ is nontrivial and is determined by the weighted 3-connected components. As a corollary, the edge-deletion problem $P_{{\text{ED}}} (\pi )$ on planar graphs is NP-complete for several properties $\pi $, such as $\pi = $ “series-parallel,” “outerplanar,” “without cocycles of cardinality $ \geqq 3$,” etc.