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Connectivity in Hypergraphs

Published online by Cambridge University Press:  20 November 2018

Megan Dewar
Affiliation:
Tutte Institute for Mathematics and Computing, Ottawa, ON, e-mail: tutte.institute+megandewar@gmail.com
David Pike
Affiliation:
Tutte Institute for Mathematics and Computing, Ottawa, ON, e-mail: tutte.institute+johnproos@gmail.com
John Proos
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, e-mail: dapike@mun.ca

Abstract

In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that, while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard, while the other can be solved in polynomial time.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

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