A large class of maximally tough graphs

Doty, Lynne L.

doi:10.1007/bf01720148

Cited by 7 publications

(3 citation statements)

References 5 publications

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“…He also conjectured that for r odd and n sufficiently large, it would be necessary that n ≡ 0 mod r, and verified this for r = 3. But for all odd r ≥ 5, Doty [78] and Jackson and Katerinis [114] independently constructed an infinite family of r-regular, r/2-tough graphs on n vertices with n ≡ 0 mod r.…”

Section: Theorem 85 1-tough Remains Np-hard For Bipartite Graphsmentioning

confidence: 99%

Toughness in Graphs – A Survey

Bauer

Broersma

Schmeichel

2006

Graphs and Combinatorics

106

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In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!

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Section: Theorem 85 1-tough Remains Np-hard For Bipartite Graphsmentioning

confidence: 99%

Toughness in Graphs – A Survey

Bauer

Broersma

Schmeichel

2006

Graphs and Combinatorics

106

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“…The toughness invariant was introduced by Chvatal [18] in the study of hamiltonian cycles. Initial partial results on the synthesis problem were independently obtained by Doty [21] and Jackson and Katerinis [31]; further progress has been made by Doty and Ferland [22,23]. Recently, Bauer et al have published a comprehensive survey on toughness [9].…”

Section: Network Vulnerability Measuresmentioning

confidence: 99%

A survey of some network reliability analysis and synthesis results

2009

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The purpose of this article is to introduce several results concerning the analysis and synthesis of reliable or invulnerable networks. First, the notion of signed reliability domination of systems is described and some applications to reliability analysis are reviewed. Then the analysis problem is considered and a brief summary of the difficulty of calculating various reliability measures is presented. Some relevant concepts in the synthesis of a most reliable network are studied. The article concludes with an introduction to a non-probabilistic approach to evaluate the vulnerability of a network.

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“…We list the results on the extreme values of these vulnerability parameters of graphs in the following table: Complete solution [14] Toughness Partial solution [8,9,10,11,13] Unknown…”

Section: Introductionmentioning

confidence: 99%

Extreme Tenacity of Graphs with Given Order and Size

Cheng

et al. 2014

J. Oper. Res. Soc. China

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Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured by various graph parameters, such as connectivity, toughness, scattering number, integrity, tenacity, rupture degree and edge-analogues of some of them. Among these parameters, the tenacity and rupture degree are two better ones to measure the stability of a network. In this paper, we consider two extremal problems on the tenacity of graphs: determine the minimum and maximum tenacity of graphs with given order and size. We give a complete solution to the first problem, while for the second one, it turns out that the problem is much more complicated than that of the minimum case. We determine the maximum tenacity of trees with given order and show the corresponding extremal graphs. The paper concludes with a discussion of a related problem on the edge vulnerability parameters of graphs.

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A large class of maximally tough graphs

Cited by 7 publications

References 5 publications

Toughness in Graphs – A Survey

Toughness in Graphs – A Survey

A survey of some network reliability analysis and synthesis results

Extreme Tenacity of Graphs with Given Order and Size

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