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Sparse Incidence Geometries and Pebble Game Algorithms

Signe Lundqvist, Tovohery Randrianarisoa, Klara Stokes, Joannes Vermant

TL;DR

This work defines sparsity and tightness of rank 2 incidence geometries, and develops an algorithm which recognises these properties, and makes this algorithm work not only for uniformhypergraphs, but for all hypergraphs.

Abstract

We generalize a sparsity condition for hypergraphs and show a result relating sparseness of hypergraphs to the decomposition of a modified incidence graph into edge-disjoint spanning forests. We also give new sparsity conditions for posets, and define an algorithm of pebble game type for recognising this sparsity.

Sparse Incidence Geometries and Pebble Game Algorithms

TL;DR

This work defines sparsity and tightness of rank 2 incidence geometries, and develops an algorithm which recognises these properties, and makes this algorithm work not only for uniformhypergraphs, but for all hypergraphs.

Abstract

We generalize a sparsity condition for hypergraphs and show a result relating sparseness of hypergraphs to the decomposition of a modified incidence graph into edge-disjoint spanning forests. We also give new sparsity conditions for posets, and define an algorithm of pebble game type for recognising this sparsity.
Paper Structure (10 sections, 8 theorems, 47 equations, 1 algorithm)

This paper contains 10 sections, 8 theorems, 47 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Sparsity conditions and algorithms of pebble game type
  4. Arboricity
  5. Sparsity for rank 2 incidence geometries
  6. Sparsity for posets and a pebble game algorithm
  7. $(K,L)$-sparsity for posets
  8. Pebble Game Algorithm.
  9. $K,L$ sparsity does not give a matroid
  10. Conclusions and open problems

Key Result

Lemma 2

If $(P,L,E(I_\lambda(\Gamma)))$ is $(k_1,k_2,l)$-sparse then $\Gamma$ is $(\lambda, k_1, k_2,l)$-sparse. Also, if $(P,L,E(I_\lambda(\Gamma)))$ is $(k_1,k_2,l)$-tight, then $\Gamma$ is $(\lambda, k_1, k_2,l)$-tight.

Theorems & Definitions (19)

  • Definition 1
  • Lemma 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • Definition 6
  • Example 7
  • Example 8
  • ...and 9 more