Bipartition constrained edge-splitting in directed graphs

Harold N. Gabow, T. Jordán

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let G=(V+s,E) be a digraph with ρ(s)=δ(s) which is k-edge-connected in V. Mader (European J. Combin. 3 (1982) 63) proved that there exists a pair vs,st of edges which can be "split off", that is, which can be replaced by a new edge vt, preserving k-edge-connectivity in V. Such a pair is called admissible. We extend this theorem by showing that for more than ρ(s)/2 edges vs there exist at least ρ(s)/2 edges st such that vs and st form an admissible pair. We apply this result to the problem of splitting off edges in G when a prespecified bipartition V=A∪B is also given and no edge can be split off with both endvertices in A or both in B. We prove that an admissible pair satisfying the bipartition constraints exists if ρ(s)≥2k+1. Based on this result we develop a polynomial algorithm which gives an almost optimal solution to the bipartition constrained edge-connectivity augmentation problem. In this problem we are given a directed graph H=(V,E), a bipartition V=A∪B and a positive integer k; the goal is to find a smallest set F of edges for which H′=(V,E∪F) is k-edge-connected and no edge of the augmenting set F has both endvertices in A or both in B. Our algorithm adds at most k edges more than the optimum.

Original languageEnglish
Pages (from-to)49-62
Number of pages14
JournalDiscrete Applied Mathematics
Volume115
Issue number1-3
DOIs
Publication statusPublished - Nov 15 2001

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Directed graphs
Directed Graph
Edge-connectivity
Polynomials
Polynomial Algorithm
Augmentation
Digraph
Optimal Solution
Integer
Theorem

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Bipartition constrained edge-splitting in directed graphs. / Gabow, Harold N.; Jordán, T.

In: Discrete Applied Mathematics, Vol. 115, No. 1-3, 15.11.2001, p. 49-62.

Research output: Contribution to journalArticle

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