Domination numbers and noncover complexes of hypergraphs

TL;DR

This paper investigates the homological properties of noncover complexes of hypergraphs, providing bounds on their Leray numbers based on domination numbers and determining their homotopy types for specific hypergraph classes.

Contribution

It introduces bounds on Leray numbers of noncover complexes using domination numbers and extends known graph results to hypergraphs, including homotopy type computations.

Findings

Upper bound on Leray numbers in terms of domination numbers

Homotopy types of noncover complexes for tight paths and cycles

Extension of graph results to hypergraph noncover complexes

Abstract

Let $\mathcal{H}$ be a hypergraph on a finite set $V$. A {\em cover} of $\mathcal{H}$ is a set of vertices that meets all edges of $\mathcal{H}$. If $W$ is not a cover of $\mathcal{H}$, then $W$ is said to be a {\em noncover} of $\mathcal{H}$. The {\em noncover complex} of $\mathcal{H}$ is the abstract simplicial complex whose faces are the noncovers of $\mathcal{H}$. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is applied to compute the homotopy type of the noncover complexes of certain uniform hypergraphs, called {\em tight paths} and {\em tight cycles}. This extends to hypergraphs known results on graphs.