A Study on Hub Sets in Hypergraphs

Abstract

The concept of hub sets in hypergraphs, previously introduced in the literature, is further explored in this article. Let \( H \) be a hypergraph. A subset \( S \subseteq V(H) \) is called a \emph{hub set} of \( H \) if for every pair of vertices \( u, v \in V(H) - S \), either \( u \) and \( v \) are adjacent in \( H \), or there exists an \( S \)-hyperpath connecting them. The \emph{hub number} of \( H \), denoted \( h(H) \), is defined as the minimum cardinality of such a hub set. In this work, the hub number is computed for various classes of hypergraphs. Additionally, the notion of vertex contraction in hypergraphs is introduced, and its influence on the hub number is investigated. Bounds on the hub number in terms of other hypergraph parameters are also established.