Menger's Theorem

Böhme, Thomas; Göring, Frank; Harant, Jochen

doi:10.1002/jgt.1001

Cited by 30 publications

(20 citation statements)

References 1 publication

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“…The proof technique used in [9], [31], [10] implicitly relies on Menger's theorem [5], which can be expressed as follows: there exists x disjoint paths between two nodes p and q if and only if x nodes must be removed to disconnect p and q.…”

Section: Our Contributionmentioning

confidence: 99%

Communicating Reliably in Multihop Dynamic Networks Despite Byzantine Failures

Maurer

Tixeuil²,

Défago

2015

2015 IEEE 34th Symposium on Reliable Distributed Systems (SRDS)

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We consider the following problem: two nodes want to reliably communicate in a dynamic multihop network where some nodes have been compromised, and may have a totally arbitrary and unpredictable behavior. These nodes are called Byzantine. We consider the two cases where cryptography is available and not available.We prove the necessary and sufficient condition (that is, the weakest possible condition) to ensure reliable communication in this context. Our proof is constructive, as we provide Byzantineresilient algorithms for reliable communication that are optimal with respect to our impossibility results.In a second part, we investigate the impact of our conditions in three case studies: participants interacting in a conference, robots moving on a grid and agents in the subway. Our simulations indicate a clear benefit of using our algorithms for reliable communication in those contexts.

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Section: Our Contributionmentioning

confidence: 99%

Communicating Reliably in Multihop Dynamic Networks Despite Byzantine Failures

Maurer

Tixeuil²,

Défago

2015

2015 IEEE 34th Symposium on Reliable Distributed Systems (SRDS)

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“…A new short proof has been given as recently as 2001 [3]. There are several equivalent formulations; see for example Dirac [14].…”

Section: Introductionmentioning

confidence: 99%

Graph-like continua, augmenting arcs, and Menger’s theorem

Thomassen

Vella

2008

Combinatorica

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We show that an adaptation of the augmenting path method for graphs proves Menger's Theorem for wide classes of topological spaces. For example, it holds for locally compact, locally connected, metric spaces, as already known. The method lends itself particularly well to another class of spaces, namely the locally arcwise connected, hereditarily locally connected, metric spaces. Finally, it applies to every space where every point can be separated from every closed set not containing it by a finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph. While closed subsets of such a space behave nicely in that they are compact and locally connected (and therefore locally arcwise connected), the general subspaces do not: They may be connected without being arcwise connected. Nevertheless, they satisfy Menger's Theorem.

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“…To assess our approach, we use the nem simulator 4 . We use it for traveling along the paths between any pair of nodes in the studied network.…”

Section: Simulation Settingsmentioning

confidence: 99%

End-to-End Key Exchange through Disjoint Paths in P2P Networks

Ahmat¹,

Magoni²,

Bissyandé

2015

ICST Transactions on Security and Safety

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International audienceDue to their inherent features, P2P networks have proven to be effective in the exchange of data between autonomous peers. Unfortunately, these networks are subject to various security threats that cannot be addressed readily since traditional security infrastructures, which are centralized, cannot be applied to them. Furthermore, communication reliability across the Internet is threatened by various attacks, including usurpation of identity, eavesdropping or traffic modification. Thus, in order to overcome these security issues and allow peers to securely exchange data, we propose a new key management scheme over P2P networks. Our approach introduces a new method that enables a secret key exchange through disjoint paths in the absence of a trusted central coordination point which would be required in traditional centralized security systems

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Menger's Theorem

Abstract: Menger's Theorem for digraphs states that for any two vertex sets A and B of a digraph D such that A cannot be separated from B by a set of at most t vertices, there are t +1 disjoint A±B-paths in D. Here a short and elementary proof of a more general theorem is given.

Cited by 30 publications

References 1 publication

Communicating Reliably in Multihop Dynamic Networks Despite Byzantine Failures

Communicating Reliably in Multihop Dynamic Networks Despite Byzantine Failures

Graph-like continua, augmenting arcs, and Menger’s theorem

End-to-End Key Exchange through Disjoint Paths in P2P Networks

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