The hardness of the independence and matching clutter of a graph

Abstract

A clutter (or antichain or 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 vertices of $L$, and the elements of $E$ are called edges of $L$. A subset se of an edge e of a clutter is called recognizing for e, if $s_e$ is not a subset of another edge. The hardness of an edge $e$ of a clutter is the ratio of the size of $e$'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.