  We introduce the concept of matching connectivity as a notion of connectivity
in graph admitting perfect matchings which heavily relies on the structural
properties of those matchings. We generalise a result of Robertson, Seymour and
Thomas for bipartite graphs with perfect matchings (see [Neil Roberts, Paul D
Seymour, and Robin Thomas. Permanents, pfaffian orientations, and even directed
curcuits. Annals of Mathematics, 150(2):929-975, 1999]) in order to obtain a
concept of alternating paths that turns out to be sufficient for the
description of our connectivity parameter. We introduce some basic properties
of matching connectivity and 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.
To be more precise we show that every n-extendable graph is matching
n-connected and for the converse every matching (n+1)-connected graph either is
n-extendable, or belongs to a well described class of graphs: the brace
h-critical graphs.
