A classification of 4-connected graphs

Slater, Peter J.

doi:10.1016/0095-8956(74)90034-3

Cited by 43 publications

(26 citation statements)

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“…A discussion of this and other reduction methods for 3-connected graphs can be found in [a]. A recursive description of the 4-connected graphs was found by Slater [52], but it is not easy to apply. A simpler and perhaps more applicable description has been announced by Martinov [34]: In a 4-connected graph G it is always possible to delete or contract an edge, such that the resulting graph is 4-connected, unless G is the line graph of a cubic cyclically 4-edgeconnected graph.…”

Section: A Graph G Has Chromatic Number At Least K If and Only Ifmentioning

confidence: 99%

Reflections on graph theory

Thomassen

1986

Journal of Graph Theory

141

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Section: A Graph G Has Chromatic Number At Least K If and Only Ifmentioning

confidence: 99%

Reflections on graph theory

Thomassen

1986

Journal of Graph Theory

141

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“…Currently, no linear-time algorithm for testing graphs on 4-connectivity is known. Interestingly, construction sequences for 4-connected graphs exist [49]. Downloaded 11/19/14 to 167.96.17.248.…”

Section: Simplificationsmentioning

confidence: 98%

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

Schmidt

2013

SIAM J. Comput.

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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

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“…Therefore by (5), (6) and (7) we have r(M * ) + 1 = r(M * ), a contradiction. This completes the proof.…”

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confidence: 91%

On dual of the generalized splitting matroids

Ghafari

Azadi

Azanchiler

2018

asat

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Given a binary matroid M and a subset T ⊆ E(M ), Luis A. Goddyn posed a problem that the dual of the splitting of M , i.e., ((MT ) * ) is not always equal to the splitting of the dual of M , ((M * )T ). This persuade us to ask if we can characterize those binary matroids for which (MT ) * = (M * )T . Santosh B. Dhotre answered this question for a two-element subset T . In this paper, we generalize his result for any subset T ⊆ E(M ) and exhibit a criterion for a binary matroid M and subsets T for which (MT ) * and (M * )T are the equal. We also show that there is no subset T ⊆ E(M ) for which, the dual of element splitting of M , i.e., ((M ′ T ) * ) equals to the element splitting of the dual of M , ((M * ) ′ T ).

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A classification of 4-connected graphs

Cited by 43 publications

References 2 publications

Reflections on graph theory

Reflections on graph theory

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

On dual of the generalized splitting matroids

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