Dependence Relation and Removable Classes

Deletions and contractions of edges are two common inductive tools in graph theory. For example, the famous theorem of Kuratowski states that a graph is nonplanar if and only if it can be reduced to either K5 or to K3,3 by means of deletions and contractions of edges. There are several theorems of similar flavour in the theory of matching covered graphs. But here the deletion and contraction operations used are, of necessity, more restrictive as the deletion of an arbitrary edge from a matching covered graph need not result in a matching covered graph, and the contraction of a single edge does not even preserve the parity of the number of vertices.