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A new axiom system for matroids: 1. Uniform matroid recognition

Brahim Chaourar

Abstract

In this paper, we give a new axioms system based on nonseparable flats with their ranks to define a matroid. We deduce a polynomial time algorithm for deciding if a given matroid (respectively, arbitrary structure) is an uniform matroid. This problem is intractable if we use an independence or an equivalent oracle.

A new axiom system for matroids: 1. Uniform matroid recognition

Abstract

In this paper, we give a new axioms system based on nonseparable flats with their ranks to define a matroid. We deduce a polynomial time algorithm for deciding if a given matroid (respectively, arbitrary structure) is an uniform matroid. This problem is intractable if we use an independence or an equivalent oracle.

Paper Structure

This paper contains 4 sections, 30 theorems.

Table of Contents

  1. Introduction
  2. $w_0$-systems are matroids
  3. Uniform matroid recognition
  4. Conclusion

Key Result

Lemma 2.1

Let $E$ be a finite set, $f$, $f_1$, $f_2$ three nonnegative integer functions defined on $2^E$, $\mathcal{E}\subseteq \mathcal{A}, \mathcal{A}_i\subseteq 2^E$, $i=1, 2$, and $\O \neq X\subseteq Y\subseteq E$. Then (i) If $\mathcal{A}_1\subseteq \mathcal{A}_2$ then $f(X, \mathcal{A}_2)\leq f(X, \mat

Theorems & Definitions (50)

  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 40 more