On the structure of spikes

TL;DR

This paper investigates the structure of spikes, a class of 3-connected matroids, and provides conditions under which the es-splitting operation can generate higher-rank spikes from lower-rank ones, revealing a recursive construction method.

Contribution

It establishes necessary and sufficient conditions for the es-splitting operation to produce higher-rank spikes directly from lower-rank spikes, enabling recursive construction of these matroids.

Findings

Characterizes when es-splitting yields $Z_{r+1}$ from $Z_{r}$.

Shows all binary and many non-binary spikes derive from $Z_{3}$.

Provides a recursive method to construct spikes via es-splitting and relaxations.

Abstract

Spikes are an important class of 3-connected matroids. For an integer $r\geq 3$, 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 $Z_{r+1}$ 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.