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The Homomorphism Poset of K_{2,n}

Sally Cockburn, Yonghyun Song

Abstract

A geometric graph is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. Two geometric realizations of a simple graph are geo-isomorphic if there is a vertex bijection between them that preserves vertex adjacencies and non-adjacencies, as well as edge crossings and non-crossings. A natural extension of graph homomorphisms, geo-homomorphisms, can be used to define a partial order on the set of geo-isomorphism classes of realizations of a given simple graph. In this paper, the homomorphism poset of the complete bipartite graph K_{2,n} is determined by establishing a correspondence between realizations of K_{2,n} and permutations of S_n, in which crossing edges correspond to inversions. Through this correspondence, geo-isomorphism defines an equivalence relation on S_n, which we call geo-equivalence. The number of geo-isomorphism classes is provided for all n <= 9. The modular decomposition tree of permutation graphs is used to prove some results on the size of geo-equivalence classes. A complete list of geo-equivalence classes and a Hasse diagrams of the poset structure are given for n <= 5.

The Homomorphism Poset of K_{2,n}

Abstract

A geometric graph is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. Two geometric realizations of a simple graph are geo-isomorphic if there is a vertex bijection between them that preserves vertex adjacencies and non-adjacencies, as well as edge crossings and non-crossings. A natural extension of graph homomorphisms, geo-homomorphisms, can be used to define a partial order on the set of geo-isomorphism classes of realizations of a given simple graph. In this paper, the homomorphism poset of the complete bipartite graph K_{2,n} is determined by establishing a correspondence between realizations of K_{2,n} and permutations of S_n, in which crossing edges correspond to inversions. Through this correspondence, geo-isomorphism defines an equivalence relation on S_n, which we call geo-equivalence. The number of geo-isomorphism classes is provided for all n <= 9. The modular decomposition tree of permutation graphs is used to prove some results on the size of geo-equivalence classes. A complete list of geo-equivalence classes and a Hasse diagrams of the poset structure are given for n <= 5.

Paper Structure

This paper contains 6 sections, 25 theorems, 54 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Permutations and Realizations of $K_{2,n}$
  3. Geo-equivalence Classes
  4. Size of Geo-equivalence Classes
  5. Poset Structure
  6. Open Questions

Key Result

Theorem 1

Let $\pi \in S_n$ and $i, j \in \{ 1, 2, \dots, n\}$. Then $bi$ crosses $aj$ in $\overline{K}_{2,n}(\pi)$ if and only if $(i,j)\in E(\pi)$.

Figures (16)

  • Figure 1: The homomorphism poset $\mathcal{K}_{2,2}$.
  • Figure 2: Template for the construction.
  • Figure 3: The realization $\overline{K}_{2,4}(2431)$.
  • Figure 4: Inversions correspond to crossings.
  • Figure 5: Re-labeling vertices of a realization of $\overline{K}_{2,8}$.
  • ...and 11 more figures

Theorems & Definitions (59)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • ...and 49 more