On the structure of spikes

Abstract

Spikes are an important class of 3-connected matroids. For an integer r ! 3, there is a unique binary r-spike denoted by Z r. When a circuit-hyperplane of Z r is relaxed, we obtain another spike and repeating this procedure will produce other non-binary spikes. The es-splitting operation on a binary spike of rank r, may not yield a spike. In this paper, we give a necessary and sufficient condition for the es-splitting operation to construct Z rþ1 directly from Z r. Indeed, all binary spikes and many of non-binary spikes of each rank can be derived from the spike Z 3 by a sequence of the es-splitting operations and circuit-hyperplane relaxations.