Tutte proved that if e is an element of a 3-connected matroid M such that neither M\e nor M/e is 3-connected, then e is in a 3-circuit or a 3-cocircuit. In this paper, we prove a broad generalization… Expand

Mathematical Proceedings of the Cambridge Philosophical Society

1981

This paper relates the concept of n-connection for graphs to Tutte's theory of n-connection for matroids (12). In particular, we show how Tutte's definition may be modified to give a matroid concept… Expand

Each of these classes of unavoidable matroids is characterized in terms of an extremal connectivity condition and it is proved that ifMis a 3-connected matroid of at least rank 7 for which every single-element deletion or contraction is 3- connected but no 2-element contraction is, thenMis a spike with its tip deleted.Expand

Definition 1. A graph is called 2-connected if it contains at least 2 vertices and each pair of vertices belong to some circuit contained in the graph. For graphs with at least 3 vertices this is… Expand

Let C1 , C2 ,· · · ,Cm be the columns of a matrix M. Any subset of these columns is either linearly independent or linearly dependent; the subsets thus fall into two classes. These classes are not… Expand