Bounding the trace function of a hypergraph with applications

TL;DR

This paper introduces a new upper bound on the trace function of hypergraphs, leading to improved methods for computing VC dimension and distinguishing transversal number, with applications in graph domination theory.

Contribution

It presents a novel upper bound on the hypergraph trace function and demonstrates its applications in efficiently estimating VC dimension and transversal numbers.

Findings

Improves polynomial-time computation of VC dimension for hypergraphs with bounded degeneracy.

Provides a new lower bound on the distinguishing transversal number.

Enhances understanding of hypergraph properties related to graph domination.

Abstract

An upper bound on the trace function of a hypergraph $H$ is derived and its applications are demonstrated. For instance, a new upper bound for the VC dimension of $H$, or $vc(H)$, follows as a consequence and can be used to compute $vc(H)$ in polynomial time provided that $H$ has bounded degeneracy. This was not previously known. Particularly, when $H$ is a hypergraph arising from closed neighborhoods of a graph, this approach asymptotically improves the time complexity of the previous result for computing $vc(H)$. Another consequence is a general lower bound on the {\it distinguishing transversal number } of $H$ that gives rise to applications in domination theory of graphs. To effectively apply the methods developed here, one needs to have good estimations of degeneracy, and its variation or reduced degeneracy which is introduced here.