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Super-minimally $3$-connected matroids

Wayne Ge, James Oxley

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.

Super-minimally $3$-connected matroids

Abstract

A super-minimally -connected matroid is a -connected matroid having no proper -connected restriction of size at least . This extends the corresponding concept for graphs. For and , we determine the maximum size of a super-minimally -connected rank- matroid and characterize, in each case, those matroids attaining the extremal bound. These results parallel Murty's results for minimally -connected matroids and Oxley's results for minimally -connected matroids.
Paper Structure (8 sections, 23 theorems, 27 equations, 2 figures, 1 table)

This paper contains 8 sections, 23 theorems, 27 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some connectivity results
  4. Small $3$-connected matroids
  5. Structural Lemmas
  6. Brittle matroids
  7. Density of super-minimally $3$-connected matroids
  8. Triads in super-minimally $3$-connected matroids

Key Result

Proposition 1.1

A matroid $M$ is super-minimally $2$-connected if and only if $M$ is isomorphic to $U_{1,1}$, or $U_{r,r+1}$ for some integer $r\geq 0$. In particular, $|E(M)|\leq r(M)+1$.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (29)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Lemma 2.8
  • ...and 19 more