Elsevier

Discrete Mathematics

Volume 307, Issue 24, 28 November 2007, Pages 3207-3212
Discrete Mathematics

Note
Local-edge-connectivity in digraphs and oriented graphs

https://doi.org/10.1016/j.disc.2007.03.051Get rights and content
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Abstract

A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivity λ(u,v) of two vertices u and v in a digraph or graph D is the maximum number of edge-disjoint uv paths in D, and the edge-connectivity of D is defined as λ(D)=min{λ(u,v)|u,vV(D);uv}. Clearly, λ(u,v)min{d+(u),d-(v)} for all pairs u and v of vertices in D. Let δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected when λ(D)=δ(D) and maximally local-edge-connected whenλ(u,v)=min{d+(u),d-(v)}for all pairs u and v of vertices in D.

In this paper we show that some known sufficient conditions that guarantee equality of λ(D) and minimum degree δ(D) for an oriented graph D also guarantee that D is maximally local-edge-connected. In addition, we generalize a result by Fàbrega and Fiol [Bipartite graphs and digraphs with maximum connectivity, Discrete Appl. Math. 69 (1996) 271–279] that each bipartite digraph of diameter at most three is maximally edge-connected.

MSC

05C40

Keywords

Local-edge-connectivity
Edge-connectivity
Minimum degree
Digraph
Oriented graph

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