On the complexity of recognizing tough graphs

Bauer, Douglas C.; Morgana, A.; Schmeichel, E. F.

doi:10.1016/0012-365x(92)00047-u

Cited by 15 publications

(16 citation statements)

References 6 publications

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“…We can use this to give a polynomial algorithm for finding an optimal k-augmentation of a (k − 1)-connected k-independence free graph. Note, however, that it is unlikely that there exists an efficient algorithm to determine b k (G) for an arbitrary graph G. This follows since the problem of determining whether b k (G) k for some 1 k |V | is NP-complete by Bauer et al [1].…”

Section: Coresmentioning

confidence: 97%

“…and r 2, we may use Theorem 3.4 to deduce without loss of generality that there is an admissible split in G + s containing sx 1 . Choose sw such that sx 1 , sw is an admissible split in G + s.…”

Section: Independence Free Graphsmentioning

confidence: 99%

“…In the former case, we apply the result of Section 4 to G+F 1 . In the latter case, we find an optimal k-augmenting set for G + F 1 using a result on 'detachments' of 2-connected graphs.…”

Section: Introductionmentioning

confidence: 97%

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Independence free graphs and vertex connectivity augmentation

Jackson

Jordán

2005

Journal of Combinatorial Theory, Series B

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Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general.In this paper, we develop an algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, our algorithm is polynomial for all k. We also derive a min-max formula for the size of a smallest augmenting set in this case. A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

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Section: Coresmentioning

confidence: 97%

Section: Independence Free Graphsmentioning

confidence: 99%

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Independence free graphs and vertex connectivity augmentation

Jackson

Jordán

2005

Journal of Combinatorial Theory, Series B

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“…Now we can easily obtain an alternative proof of Theorem by using the following result from , since

| V (G) | - c_{0} \geq (\frac{t}{t + 1}) | V (G) |

if we assume

| V (G) | \geq c_{0} (t + 1)

. Theorem Let

t \geq 1

.…”

Section: Properties Of Scriptfa(mlr)‐free Graphsmentioning

confidence: 98%

Forbidden Induced Subgraphs for Toughness

Ota¹,

Sueiro²

2012

J. Graph Theory

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Let scriptF be a family of connected graphs. A graph G is said to be scriptF‐free if G is H‐free for every graph H in scriptF. We study the relation between forbidden subgraphs in a connected graph G and the resulting toughness of G. In particular, we consider the problem of characterizing the graph families scriptF such that every large enough connected scriptF‐free graph is t‐tough. In this article, we solve this problem for every real positive number t. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 191–202, 2013

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“…Unfortunately, this measure is difficult to calculate: As shown by Bauer et al [3], it is NP-hard to determine the toughness of a graph.…”

Section: Introductionmentioning

confidence: 98%

Various results on the toughness of graphs

Broersma¹,

Engbers²,

Trommel³

1999

Networks

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Let G be a graph and let t ≥ 0 be a real number. Then, G is t‐tough if tω(G − S) ≤ |S| for all S ⊆ V(G) with ω(G − S) > 1, where ω(G − S) denotes the number of components of G − S. The toughness of G, denoted by τ(G), is the maximum value of t for which G is t‐tough [taking τ(Kn) = ∞ for all n ≥ 1]. G is minimally t‐tough if τ(G) = t and τ(H) < t for every proper spanning subgraph H of G. We discuss how the toughness of (spanning) subgraphs of G and related graphs depends on τ(G), we give some sufficient degree conditions implying that τ(G) ≥ t, and we study which subdivisions of 2‐connected graphs have minimally 2‐tough squares. © 1999 John Wiley & Sons, Inc. Networks 33: 233–238, 1999

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On the complexity of recognizing tough graphs

Cited by 15 publications

References 6 publications

Independence free graphs and vertex connectivity augmentation

Independence free graphs and vertex connectivity augmentation

Forbidden Induced Subgraphs for Toughness

Various results on the toughness of graphs

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