Super-minimally $3$-connected matroids

TL;DR

This paper investigates super-minimally k-connected matroids, extending graph concepts, and determines maximum sizes and characterizations for such matroids when k=2 and 3, paralleling known graph and matroid results.

Contribution

It extends the concept of minimal connectivity to super-minimal cases in matroids and characterizes extremal structures for k=2 and 3.

Findings

Maximum size of super-minimally 2-connected matroids determined.

Maximum size of super-minimally 3-connected matroids determined.

Characterizations of extremal super-minimally 2- and 3-connected matroids provided.

Abstract

A super-minimally $k$-connected matroid is a $k$-connected matroid having no proper $k$-connected restriction of size at least $2k-2$. This extends the corresponding concept for graphs. For $k=2$ and $k=3$, we determine the maximum size of a super-minimally $k$-connected rank-$r$ matroid and characterize, in each case, those matroids attaining the extremal bound. These results parallel Murty's results for minimally $2$-connected matroids and Oxley's results for minimally $3$-connected matroids.