The hardness of the independence and matching clutter of a graph

TL;DR

This paper investigates the computational complexity of determining the hardness of clutters derived from graph independent sets and matchings, revealing insights into their structural properties and algorithmic challenges.

Contribution

It introduces the concept of hardness for graph-based clutters and analyzes its computational complexity, providing new theoretical understanding of these combinatorial structures.

Findings

Hardness of independent set clutters is computationally complex.

Hardness of matching clutters exhibits similar complexity challenges.

Provides bounds and properties related to the hardness measure.

Abstract

A {\it clutter} (or {\it antichain} or {\it Sperner family}) $L$ is a pair $(V,E)$, where $V$ is a finite set and $E$ is a family of subsets of $V$ none of which is a subset of another. Usually, the elements of $V$ are called {\it vertices} of $L$, and the elements of $E$ are called {\it edges} of $L$. A subset $s_e$ of an edge $e$ of a clutter is called {\it recognizing} for $e$, if $s_e$ is not a subset of another edge. The {\it hardness} of an edge $e$ of a clutter is the ratio of the size of $e\textrm{'s}$ smallest recognizing subset to the size of $e$. The hardness of a clutter is the maximum hardness of its edges. We study the hardness of clutters arising from independent sets and matchings of graphs.