Abstract
The notion of removable classes arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. The last (single or double) ear of an ear decomposition is defined as a removable class. Every matching covered graph not induced by a circuit has at least three removable classes. In this paper, we 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. Let $G$ be a matching covered graph. A matching covered subgraph $H$ of $G$ is conformal if $G-VH$ has a perfect matching. Given $S \subseteq EG$, what is a minimal conformal subgraph of $G$ that contains $S$? It is known that if $|S|=2$ then it is induced by a circuit. As an application of the main result, we answer this question for $|S|=3$.
Highlights
The graphs considered here are loopless, but they may have multiple edges
The notion of removable classes arises in connection with ear decompositions of matching covered graphs introduced by Lovasz and Plummer
Every matching covered graph not induced by a circuit has at least three removable classes
Summary
The graphs considered here are loopless, but they may have multiple edges. The notation and terminology we use is essentially that of Bondy and Murty [1]. A connected graph G is matching covered if each of its edges lies in a perfect matching. Some authors refer to matching covered graphs as 1-extendable graphs. The treatise by Lovasz and Plummer [11] and the seminal work by Lovasz [10] on the matching lattice contain the basic theory of matching covered graphs. For the convenience of the reader, we shall briefly review here the terminology and results which are pertinent to this article
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