A proof of Menger's theorem

1969 ◽  
Vol 73 (1) ◽  
pp. 81-83 ◽  
Author(s):  
J. S. Pym
Keyword(s):  
Menger's Theorem

A remark on Menger's theorem

1970 ◽  
Vol 21 (3-4) ◽  
pp. 365-368 ◽  
Author(s):  
L. Lovász
Keyword(s):  
Menger's Theorem

Structure and Substructure Connectivity of Hypercube-Like Networks

2020 ◽  
Vol 30 (03) ◽  
pp. 2040007
Author(s):  
Cheng-Kuan Lin ◽  
Eddie Cheng ◽  
László Lipták
Keyword(s):  
Text Structure ◽  
Network Architectures ◽  
Connected Subgraph ◽  
Menger's Theorem ◽  
The Family ◽  
Minimum Number ◽  
Structure Formula ◽  
Connected Subgraphs ◽  
Twisted Cubes

The connectivity of a graph [Formula: see text], [Formula: see text], is the minimum number of vertices whose removal disconnects [Formula: see text], and the value of [Formula: see text] can be determined using Menger’s theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. Two extensions to the classic notion of connectivity were introduced recently: structure connectivity and substructure connectivity. Let [Formula: see text] be isomorphic to any connected subgraph of [Formula: see text]. The [Formula: see text]-structure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] such that every element of [Formula: see text] is isomorphic to [Formula: see text], and the removal of [Formula: see text] disconnects [Formula: see text]. The [Formula: see text]-substructure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] whose removal disconnects [Formula: see text] and every element of [Formula: see text] is isomorphic to a connected subgraph of [Formula: see text]. The family of hypercube-like networks includes many well-defined network architectures, such as hypercubes, crossed cubes, twisted cubes, and so on. In this paper, both the structure and substructure connectivity of hypercube-like networks are studied with respect to the [Formula: see text]-star [Formula: see text] structure, [Formula: see text], and the [Formula: see text]-cycle [Formula: see text] structure. Moreover, we consider the relationships between these parameters and other concepts.

scholarly journals A link between menger’s theorem and infinite euler graphs

1997 ◽  
Vol 74 (1-2) ◽  
pp. 41-46
Author(s):  
L. Boza ◽  
A. Diánez ◽  
A. Márquez ◽  
A. Quintero
Keyword(s):  
Menger's Theorem

scholarly journals Menger's theorem for infinite graphs with ends

2005 ◽  
Vol 50 (3) ◽  
pp. 199-211 ◽  
Author(s):  
Henning Bruhn ◽  
Reinhard Diestel ◽  
Maya Stein
Keyword(s):  
Infinite Graphs ◽  
Menger's Theorem

scholarly journals Countable Menger's Theorem with Finitary Matroid Constraints on the Ingoing Edges

10.37236/6586 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Attila Joó
Keyword(s):  
Edge Disjoint ◽  
Menger's Theorem

We present a strengthening of the countable Menger's theorem of R. Aharoni. Let $ D=(V,A) $ be a countable digraph with $ s\neq t\in V $ and let $\mathcal{M}=\bigoplus_{v\in V}\mathcal{M}_v $ be a matroid on $ A $ where $ \mathcal{M}_v $ is a finitary matroid on the ingoing edges of $ v $. We show that there is a system of edge-disjoint $ s \rightarrow t $ paths $ \mathcal{P} $ such that the united edge set of these paths is $ \mathcal{M} $-independent, and there is a $ C \subseteq A $ consisting of one edge from each element of $\mathcal{P} $ for which $ \mathsf{span}_{\mathcal{M}}(C) $ covers all the $ s\rightarrow t $ paths in $ D $.

Menger's Theorem and Matroids

1971 ◽  
Vol s2-4 (1) ◽  
pp. 46-50 ◽  
Author(s):  
R. A. Brualdi
Keyword(s):  
Menger's Theorem
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