The Homomorphism Poset of K_{2,n}

TL;DR

This paper characterizes the structure of the homomorphism poset of the complete bipartite graph K_{2,n} by relating geometric realizations to permutations, providing classifications and visualizations for small n.

Contribution

It establishes a correspondence between geometric realizations of K_{2,n} and permutations, defining geo-equivalence, and determines the poset structure for small n.

Findings

Number of geo-isomorphism classes for n <= 9

Complete list of classes for n <= 5

Hasse diagrams illustrating the poset structure

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.