Matching Connectivity: On the Structure of Graphs with Perfect Matchings
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.
Keywords
Cite
@article{arxiv.1704.00493,
title = {Matching Connectivity: On the Structure of Graphs with Perfect Matchings},
author = {Archontia C. Giannopoulou and Stephan Kreutzer and Sebastian Wiederrecht},
journal= {arXiv preprint arXiv:1704.00493},
year = {2019}
}
Comments
Error in main proof