# Ear Decompositions of Matching Covered Graphs

@article{Carvalho1999EarDO,
title={Ear Decompositions of Matching Covered Graphs},
author={Marcelo H. de Carvalho and Claudio L. Lucchesi and Uppaluri S. R. Murty},
journal={Combinatorica},
year={1999},
volume={19},
pages={151-174}
}
• Published 1 February 1999
• Mathematics
• Combinatorica
G different from and has at least Δ edge-disjoint removable ears, where Δ is the maximum degree of G. This shows that any matching covered graph G has at least Δ! different ear decompositions, and thus is a generalization of the fundamental theorem of Lovász and Plummer establishing the existence of ear decompositions. We also show that every brick G different from and has Δ− 2 edges, each of which is a removable edge in G, that is, an edge whose deletion from G results in a matching covered…
Brick Generation and Conformal Subgraphs
This theorem is a refinement of the result of Carvalho, Lucchesi and Murty (2003) which is appropriate for the restricted class of near-bipartite bricks and establishes generation theorems which are specific to near-magnifying bricks.
On a Conjecture of Lovász Concerning Bricks: I. The Characteristic of a Matching Covered Graph
• Mathematics
J. Comb. Theory, Ser. B
• 2002
The notion of the characteristic of a separating cut in a matching covered graph is introduced and some basic properties are established to first prove the theorem for solid bricks, that is, bricks which do not have any nontrivial separating cuts.
Matching Covered Graphs with Three Removable Classes
• Mathematics
Electron. J. Comb.
• 2014
This paper characterize matching covered graphs with precisely three removable classes and show, as a corollary, that every non-planar matching covered graph has at least four removable classes.
On perfect matchings in matching covered graphs
• Mathematics
J. Graph Theory
• 2019
There exist infinitely many $k-regular graphs of class 1 with an arbitrarily large equivalent class$K$such that K is not switching-equivalent to either$\emptyset$or$E(G)$, which provides a negative answer to the problem proposed by Lukot'ka and Rollova. K4‐free and C6¯ ‐free Planar Matching Covered Graphs • Mathematics J. Graph Theory • 2016 A solution to the problem of deciding which matching covered graphs are K4-based and which are -based is presented in the special case of planar graphs and the principal tool for proving the results is the brick generation procedure established by Norine and Thomas. König Deletion Sets and Vertex Covers above the Matching Size • Mathematics ISAAC • 2008 It is shown that the problem of deleting at most k vertices to make a given graph Konig-Egervary is fixed-parameter tractable with respect to k and an interesting parameter-preserving reduction from the vertex-deletion version of red/blue-split graphs to a version of Vertex Cover is shown. The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover • Mathematics, Computer Science Algorithmica • 2010 While studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, a number of interesting structural results on matchings and vertex covers are shown which could be useful in other contexts. Ear decomposition and induced even cycles • Mathematics Discret. Appl. Math. • 2019 On the number of dissimilar pfaffian orientations of graphs • Mathematics RAIRO Theor. Informatics Appl. • 2005 It is deduced that the problem of determining whether or not a graph is Pfaffian is as difficult as the problemof determining whether a given orientation is PfAffian, a result first proved by Vazirani and Yanakakis. Even cycles and perfect matchings in claw-free plane graphs • Mathematics Discret. Math. Theor. Comput. Sci. • 2020 It is shown that the only cycle-nice simple 3-connected claw-free plane graphs are$K_4$,$W_5$and$\overline C_6$and every cycle- Nice 2- connected claw- free plane graph can be obtained from a graph in the family${\cal F}\$ by a sequence of three types of operations.

## References

SHOWING 1-10 OF 15 REFERENCES
Matching structure and the matching lattice
Ear-decompositions of matching-covered graphs
It is proved that a non-bipartite matching-covered graph contains K4 or K2⊕K3 (the triangular prism).
Matching Covered Graphs and Subdivisions of K 4 and C 6
We give a very simple proof that every non-bipartite matching covered graph contains a nice subgraph that is an odd subdivision of K 4 or C 6. It follows immediately that every brick diierent from K
The Factorization of Linear Graphs
I . E . Borel, Lefons sur lea series divergentes (Paris, 1901), 164 et seq. 2. P. Dienes, The Taylor Series (Oxford, 1931). 3 . Y. Okada, " Uber die Annaherung analytischer Funktionen", Math.
On Representatives of Subsets
Let a set S of mn things be divided into m classes of n things each in two distinct ways, (a) and (b); so that there are m (a)-classes and m (b)-classes. Then it is always possible to find a set R of
Decomposição otima em orelhas para grafos matching covered
• Physics
• 1996
O assunto do qual se trata este trabalho se insere na area de teoria dos grafos, e mais especificamente de grafos matching covered, que sao grafos conexos em que toda aresta pertence a um
Matching covered graphs and odd subdivi- sions of K4 and C6
• J. Combinatorial Theory (B),
• 1996
An algorithm for the ear decomposition of a 1-factor covered graph
• Mathematics
Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics
• 1989
Abstract We give a constructive proof for the theorem of Lovász and Plummer which asserts the existence of an ear decomposition of a 1-factor covered graph.
The Two Ear Theorem on Matching-Covered Graphs
We give a simple and short proof for the two ear theorem on matching-covered graphs which is a well-known result of Lovasz and Plummer. The proof relies only on the classical results of Tutte and
The Matching Lattice and Related Topics
• The Matching Lattice and Related Topics
• 1994