Matching critical intersection hypergraphs

A matching in a hypergraph H is a set of pairwise vertex disjoint edges in H and the matching number of H is the maximum cardinality of a matching in H. A hypergraph H is an intersecting hypergraph if every two distinct edges of H have a non-empty intersection. Equivalently, H is an intersecting hypergraph if and only if it has matching number one. In this paper, we study intersecting hypergraphs that are matching critical in the sense that the matching number increases under various definitions of criticality.