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Simplification & Incidence: How an Incidence-focused Perspective Patches Category-theoretic Problems in Graph Theory

Will Grilliette

Abstract

By applying simplification operations to categories of multigraphs, several natural graph operations are shown to demonstrate categorical issues. The replacement of an undirected edge with a directed cycle for digraphs admits both a left and a right adjoint, while the analogous operation for quivers only admits a left adjoint. The clique-replacement graph, intersection graph, and dual hypergraph fail to be functorial with traditional graph homomorphisms. The three failures are remedied by considering weak set-system homomorphisms, which form a category isomorphic to both the category of incidence structures and a lax comma category.

Simplification & Incidence: How an Incidence-focused Perspective Patches Category-theoretic Problems in Graph Theory

Abstract

By applying simplification operations to categories of multigraphs, several natural graph operations are shown to demonstrate categorical issues. The replacement of an undirected edge with a directed cycle for digraphs admits both a left and a right adjoint, while the analogous operation for quivers only admits a left adjoint. The clique-replacement graph, intersection graph, and dual hypergraph fail to be functorial with traditional graph homomorphisms. The three failures are remedied by considering weak set-system homomorphisms, which form a category isomorphic to both the category of incidence structures and a lax comma category.
Paper Structure (17 sections, 18 theorems, 20 equations, 3 figures)

This paper contains 17 sections, 18 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Category of Sets
  4. Category of Quasi-ordered Sets
  5. Lax Comma Category
  6. The Fallacy of Intuition
  7. A Diagram of Classical Graph Theory
  8. Quivers & Digraphs
  9. Set Systems & Hypergraphs
  10. Multigraphs & Simple Graphs
  11. Clique Replacement & Intersection Graphs
  12. Dual Hypergraph
  13. An Incidence View of Homomorphisms
  14. Incidence Structures & Hypergraphs
  15. Weak Set-system Homomorphisms
  16. ...and 2 more sections

Key Result

Theorem 2.2

If $\xymatrix{N_{\mathbf{SSys}}\left(H'\right)\ar[r]^(0.65){f} & H}\in\mathbf{SSys}$, there is a unique $\xymatrix{H'\ar[r]^(0.4){\hat{f}} & \mathop{\mathrm{Del}}\nolimits_{\mathcal{S}}(H)}\in\mathbf{Gra}$ such that $j_{H}\circ N_{\mathbf{SSys}}\left(\hat{f}\right)=f$.

Figures (3)

  • Figure 1: Diagram of Classical Graph Theory
  • Figure 2: Set-system & Incidence Hypergraphs
  • Figure 3: Preservation of Vertices and Edges

Theorems & Definitions (46)

  • Definition 1.1: Lax comma category
  • Example 1.2: Weighted sets
  • Example 1.3: Metric spaces
  • Definition 2.1: Simple deletion
  • Theorem 2.2: Characterization of $\mathop{\mathrm{Del}}\nolimits_{\mathcal{S}}$
  • Definition 2.3: Multigraph simplification
  • Theorem 2.4: Characterization of $\mathsf{S}_{\mathfrak{M}}$
  • Definition 2.5: Symmetric closure, howtoproveit
  • Theorem 2.6: Characterization of $N^{\diamond}_{\mathbf{Digra}}$
  • Definition 2.7: Symmetric interior, neuzerling2016
  • ...and 36 more