Detachments preserving local edge-connectivity of graphs

Tibor Jordán, Zoltán Szigeti

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Let G = (V + s, E) be a graph with a designated vertex s of degree d(s), and let f(s) = (d1, d2, ..., dp) be a partition of d(s) into p positive integers. An f(s)-detachment of G is a graph G′ obtained by "splitting" s into p vertices, called the pieces of s, such that the degrees of the pieces of s in G′ are given by f(s). Thus every edge sw ∈ E corresponds to an edge of G′ connecting some piece of s to w. We give necessary and sufficient conditions for the existence of an f(s)-detachment of G in which the local edge-connectivities between pairs of vertices in V satisfy prespecified lower bounds. Our result is a common generalization of a theorem of Mader on edge splittings preserving local edge-connectivities and a result of Fleiner on f(s)-detachments satisfying uniform lower bounds. It implies a conjecture of Fleiner on f(s)-detachments preserving local edge-connectivities. By using our characterization we extend a theorem of Frank on local edge-connectivity augmentation of graphs to the case when stars of given degrees are added, and we also solve the local edge-connectivity augmentation problem for 3-uniform hypergraphs.

Original languageEnglish
Pages (from-to)72-87
Number of pages16
JournalSIAM Journal on Discrete Mathematics
Volume17
Issue number1
DOIs
Publication statusPublished - Sep 1 2003

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Keywords

  • Detachments of graphs
  • Edge splitting
  • Edge-connectivity

ASJC Scopus subject areas

  • Mathematics(all)

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