Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice

Ear-decompositions of matching covered graphs

We call a graph matching-covered if every line belongs to a perfect matching. We study the technique of "ear-decompositions" of such graphs. We prove that a non-bipartite matchingcovered graph contains K~ or K2@Ka (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of r-criti...

Bases for the Matching Lattice of Matching Covered Graphs

The Two Ear Theorem on Matching-Covered Graphs

We give a simple and short proof for the two ear theorem on matchingcovered graphs which is a well-known result of Lovász and Plummer. The proof relies only on the classical results of Tutte and Hall on the existence of perfect matching in (bipartite) graphs.

Ear-decompositions and the complexity of the matching polytope

The complexity of the matching polytope of graphs may be measured with the maximum length β of a starting sequence of odd ears in an ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its facets are defined by 2-connected factor-critical graphs, which have an odd ear-decomposition (according to a theorem of Lovász). In particular, β(G) ≤ 1 if and only if the matching pol...

On Generalizations of Matching-covered Graphs

Structural results for extensions of matching-covered graphs are presented in this paper.