Matching Connectivity: On the Structure of Graphs with Perfect Matchings

TL;DR

This paper introduces matching connectivity as a new graph connectivity concept based on perfect matchings, generalizes existing results, and explores its properties and implications for n-extendable graphs.

Contribution

It generalizes a key bipartite graph result to broader classes, defines matching connectivity, and establishes its relationship with n-extendable graphs.

Findings

Proves a Menger-type theorem for matching n-connected graphs.

Shows every n-extendable graph is matching n-connected.

Characterizes graphs that are (n+1)-matching connected.

Abstract

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.