Matching Connectivity: On the Structure of Graphs with Perfect Matchings

We introduce the concept of matching connectivity as a notion of connectivity in graphs admitting perfect matchings. The notion relies heavily on structural properties of those matchings. We prove a Menger-type result for matching n-connected graphs. Furthermore, we show that matching connectivity fills a gap in the investigation of n-extendable graphs and their connectivity properties. In particular, we show that every n-extendable graph is matching n-connected and for the converse any matching (n + 1)-connected graph either is n-extendable, or belongs to a well described class of graphs: the brace h-critical graphs.