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Topological Structures of Sets and their Subsets

Fei Ma, Bing Yao

TL;DR

This paper develops a topological framework for hypergraphs by leveraging the natural topological structure of finite sets and their subsets to visualize and analyze both ordinary and non-ordinary hypergraphs. It introduces a visualization approach based on ve-intersected graphs and the Topcode-matrix to encode total colorings, constraints, and hypergraph relations, and formalizes 3I-hyperedge sets with fixed-point, strong, and additive-set structures. It then categorizes non-ordinary hypergraphs into topology-, code-, and operation-hypergraphs, detailing constructions such as $K_{tree}$- and $F_{orest}$-hypergraphs, $E_{hami}$-hypergraphs, and $G$-hypergraphs, as well as code- and parameterized-hypergraphs with diverse coloring schemes and homomorphism notions. The framework provides a rich toolkit for visualization, coloring theory, and structural analysis with potential applications in secure computation, data modeling, and topology-aware graph processing.

Abstract

For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating hypergraphs by means of the natural topological structure of finite sets and their subsets, so we are able to construct various non-ordinary hypergraphs, and to reveal topological properties (such as hamiltonian cycles, maximal planar graphs), colorings, connectivity, hypergraph group, isomorphism and homomorphism of hypergraphs.

Topological Structures of Sets and their Subsets

TL;DR

This paper develops a topological framework for hypergraphs by leveraging the natural topological structure of finite sets and their subsets to visualize and analyze both ordinary and non-ordinary hypergraphs. It introduces a visualization approach based on ve-intersected graphs and the Topcode-matrix to encode total colorings, constraints, and hypergraph relations, and formalizes 3I-hyperedge sets with fixed-point, strong, and additive-set structures. It then categorizes non-ordinary hypergraphs into topology-, code-, and operation-hypergraphs, detailing constructions such as - and -hypergraphs, -hypergraphs, and -hypergraphs, as well as code- and parameterized-hypergraphs with diverse coloring schemes and homomorphism notions. The framework provides a rich toolkit for visualization, coloring theory, and structural analysis with potential applications in secure computation, data modeling, and topology-aware graph processing.

Abstract

For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating hypergraphs by means of the natural topological structure of finite sets and their subsets, so we are able to construct various non-ordinary hypergraphs, and to reveal topological properties (such as hamiltonian cycles, maximal planar graphs), colorings, connectivity, hypergraph group, isomorphism and homomorphism of hypergraphs.

Paper Structure

This paper contains 38 sections, 47 theorems, 77 equations, 12 figures.

Table of Contents

  1. Introduction And Preliminary
  2. Researching background
  3. Hypergraphs are combinatorics of finite sets
  4. Applications of hypergraphs
  5. Preliminary
  6. Basic terminology
  7. Graph operations
  8. Basic graph colorings
  9. Hypergraphs
  10. Concepts of hypergraphs
  11. Hypergraph sets
  12. Intersected hyperedge sets
  13. Hyperedge set matchings and fixed-point hyperedge sets
  14. Strong hyperedge sets
  15. Set-set additive groups
  16. ...and 23 more sections

Key Result

Proposition 1

For a finite set $\Lambda=\{x_1,x_2,\dots ,x_m\}$, we have (i) There are particular hyperedge sets $\mathcal{E}^*_0=\Lambda$, $\mathcal{E}^*_1= \{\{x_1\}$, $\{x_2\}$, $\dots$, $\{x_m\} \}\subset \Lambda^2$ and $\mathcal{E}_r= \{\{x_r\} \}$ with $r\in [1,m]$, such that the 3I-hyperedge set $\mathcal{

Figures (12)

  • Figure 1: A scheme for illustrating the vertex-splitting operation and the vertex-coinciding operation defined in Definition \ref{['defn:vertex-split-coinciding-operations']}.
  • Figure 2: An example from an $8$-uniform hypergraph $H_{yper}$ to a ve-intersected graph $G$ defined in Definition \ref{['defn:vertex-intersected-graph-hypergraph']} admitting a set-coloring, where (a) Venn's four-set diagram using four ellipses.
  • Figure 3: A scheme for illustrating Definition \ref{['defn:Wang-Hyper-cycle']} and Definition \ref{['defn:yao-hamilton-hypergraphs']} by the ve-intersected graphs defined in Definition \ref{['defn:new-intersected-hypergraphss']}.
  • Figure 4: The set-colored graphs for Definition \ref{['defn:new-intersected-hypergraphss']} and Definition \ref{['defn:vertex-intersected-graph-hypergraph']}.
  • Figure 5: Three different set-colorings of the complete graph $K_4$.
  • ...and 7 more figures

Theorems & Definitions (161)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Definition 6
  • Remark 2
  • Definition 7
  • Example 1
  • ...and 151 more