Contractions, Removals, and Certifying 3-Connectivity in Linear Time

Abstract

One of the most noted construction methods of 3-vertex-connected graphs is due to Tutte and is based on the following fact: Any 3-vertex-connected graph G = (V, E) on more than 4 vertices contains a contractible edge, i.e., an edge whose contraction generates a 3-connected graph. This implies the existence of a sequence of edge contractions from G to the complete graph K 4 , such that every intermediate graph is 3-vertex-connected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of O(|V | 2 ) to O(|E|). This result has a number of consequences; an important one is a new linear-time test of 3-connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in recent years. The test is conceptually different from well-known linear-time 3-connectivity tests and uses a certificate that is easy to verify in time O(|E|). We show how to extend the results to an optimal certifying test of 3-edge-connectivity.1. Introduction. The class of 3-connected (i.e., 3-vertex-connected) graphs has been studied intensively for many reasons in the past 50 years. Algorithmic applications include problems in graph drawing (see [41] for a survey), problems related to planarity [8,27], and online problems on planar graphs (see [7] for a survey). From a complexity point of view, 3-connectivity is in particular important for problems dealing with longest paths, because it lies, somewhat surprisingly, on the borderline of NP-hardness: Finding a Hamiltonian cycle is NP-hard for 3-connected planar graphs [24] but becomes solvable in linear running time [13] for higher connectivity, as 4-connected planar graphs are Hamiltonian [62].From a top-level view, we design an efficient algorithm from an inductively defined construction of a graph class. For a given graph class C, such constructions start with a set of base graphs and apply iteratively operations from a finite set of operations such that precisely the members of C are constructed. Not only does this give a computational approach to test graphs for membership in C, but it can also be exploited to prove properties of C using just arguments on the base graphs and the finitely many operations. Graph theory provides inductively defined constructions for many graph classes, including planar graphs, triangulations, k-connected graphs for k ≤ 4, regular graphs, and various intersections of these classes [4,5,31]. Most of these constructions have not been exploited computationally.For an inductively defined construction of C and a graph G ∈ C, we call a sequence of operations that is applied to a specified base graph to construct G a construction sequence of G. We will also identify a construction sequence with the sequence of

λ > 4

“…While providing a certificate in the negative case is easy, defining an easy-to-check certificate in the positive case and finding such a certificate in linear time has required graphtheoretic and algorithmic innovations. 28 This example illustrates that certifiability is not primarily an issue of running time. All algorithms, for constructing, as well as for checking, the certificate, run in linear time, just like classical non-certifying algorithms.…”

Certifying 3-Edge-Connectivity

“…Our algorithm is path-based [7]. It uses the concept of a chain decomposition of a graph introduced in [25] and used for certifying 1-and 2-vertex and 2-edge-connectivity in [27] and for certifying 3-vertex connectivity in [26]. A chain decomposition is a special ear decomposition [14].…”

“…Thus, chain decompositions form a common framework for certifying k-vertexand k-edge-connectivity for k ≤ 3 in linear time. We use many techniques from [26], but in a simpler form. Hence our paper may also be used as a gentle introduction to the 3-vertex-connectivity algorithm in [26].…”

“…We use many techniques from [26], but in a simpler form. Hence our paper may also be used as a gentle introduction to the 3-vertex-connectivity algorithm in [26].…”

“…An extension of our algorithm computes the 3-edgeconnected components and the cactus representation directly (Section 11). A similar technique can be used to extend the 3-vertex-connectivity algorithm in [26] to an algorithm for computing 3-vertex-connected components.…”