The Bounds of the Edge Number in Generalized Hypertrees

Abstract

A hypergraph H = (V, ε) is a pair consisting of a vertex set V, and a set ε of subsets (the hyperedges of H) of V. A hypergraph H is r-uniform if all the hyperedges of H have the same cardinality r. Let H be an r-uniform hypergraph, we generalize the concept of trees for r-uniform hypergraphs. We say that an r-uniform hypergraph H is a generalized hypertree (GHT) if H is disconnected after removing any hyperedge E, and the number of components of GHT − E is a fixed value k (2 ≤ k ≤ r). We focus on the case that GHT − E has exactly two components. An edge-minimal GHT is a GHT whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r-uniform GHT on n vertices has at least 2n/(r + 1) edges and it has at most n − r + 1 edges if r ≥ 3 and n ≥ 3, and the lower and upper bounds on the edge number are sharp. We then discuss the case that GHT − E has exactly k (2 ≤ k ≤ r − 1) components.

A Measure for the Vulnerability of Uniform Hypergraph Networks: Scattering Number

“…While numerous studies have delved into these parameters, they predominantly focus on networks modeled by ordinary graphs, leaving a research gap in the vulnerability of hypernetworks. Currently, the literature related to vulnerability parameters in hypergraphs can be found in [15][16][17][18]. In this context, we extend the application of the scattering number as a parameter to measure the vulnerability of hypergraphs.…”

Graph-Theoretic Problems and Their New Applications

“…Zhang et al [2] consider so-called generalized hypergraphs H denoted as r-uniform if all the hyperedges have the same cardinality r. Such a graph is called a generalized hypertree GHT, if after removing any hyperedge E, GHT − E has exactly k components with 2 ≤ k ≤ r. Focusing first on the case k = 2, they determine bounds on the number of edges. In particular, the authors show that an r-uniform generalized GHT on n vertices has at least 2n/(r + 1) edges and at most n − r + 1 edges if r ≥ 3, n ≥ 3 and that the lower and upper bounds on the edge number are tight.…”