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Matroids that classify forests

Lorenzo Traldi

TL;DR

The paper addresses whether forests can be classified up to isomorphism by binary matroids derived from their adjacency matrices, introducing the restricted isotropic matroid $M[IA(F)]$ and the isotropic matroid $M[IAS(F)]$ over $GF(2)$. It proves, via self-contained linear-algebraic arguments, that for forests $F$ and $F'$, isomorphism of these matroids iff the forests are isomorphic, with Part 1 establishing $2\Rightarrow 1$ and Part 2 establishing $3\Rightarrow 1$; these results connect to local complementation and pivoting in the broader literature. The work also provides counterexamples showing the results do not generalize to arbitrary graphs and develops a nuanced view of triangulations and automorphisms (ps-equivalence) in isotropic matroids, clarifying when matroid isomorphisms reflect underlying graph structure. Overall, the paper furnishes a robust, elementary proof that forests are classifiable by the associated binary matroids, strengthening the bridge between graph theory and binary matroid theory and highlighting the boundaries of this classification in non-forest cases.

Abstract

Elementary arguments show that a tree or forest is determined (up to isomorphism) by binary matroids defined using the adjacency matrix.

Matroids that classify forests

TL;DR

The paper addresses whether forests can be classified up to isomorphism by binary matroids derived from their adjacency matrices, introducing the restricted isotropic matroid and the isotropic matroid over . It proves, via self-contained linear-algebraic arguments, that for forests and , isomorphism of these matroids iff the forests are isomorphic, with Part 1 establishing and Part 2 establishing ; these results connect to local complementation and pivoting in the broader literature. The work also provides counterexamples showing the results do not generalize to arbitrary graphs and develops a nuanced view of triangulations and automorphisms (ps-equivalence) in isotropic matroids, clarifying when matroid isomorphisms reflect underlying graph structure. Overall, the paper furnishes a robust, elementary proof that forests are classifiable by the associated binary matroids, strengthening the bridge between graph theory and binary matroid theory and highlighting the boundaries of this classification in non-forest cases.

Abstract

Elementary arguments show that a tree or forest is determined (up to isomorphism) by binary matroids defined using the adjacency matrix.

Paper Structure

This paper contains 9 sections, 20 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Binary matroids
  3. Binary matroids
  4. Isotropic matroids of forests
  5. Examples
  6. Graphs that are not forests
  7. Strange isomorphisms of isotropic matroids
  8. Part 1 of the proof
  9. Part 2 of the proof

Key Result

Theorem 3

Isotropic matroids and restricted isotropic matroids are classifying invariants for forests. That is, if $F$ and $F'$ are forests then any one of these three statements implies the other two:

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 31 more