Connectivity in Hypergraphs

TL;DR

This paper explores connectivity notions in hypergraphs, extending Whitney's theorem, and analyzes the computational complexity of finding minimum cuts, revealing polynomial-time and NP-hard cases.

Contribution

It extends Whitney's connectivity theorem to hypergraphs and characterizes the complexity of minimum cut problems in this context.

Findings

Strong vertex connectivity is bounded by weak edge connectivity.

Minimum weak vertex cut can be found in polynomial time.

Minimum strong vertex cut is NP-hard, even for hypergraphs with edges of size at most 3.

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.