The Homomorphism Poset of $K_{3,3}$

TL;DR

This paper characterizes the structure of the homomorphism poset of the complete bipartite graph K_{3,3} by analyzing geometric realizations and their homomorphisms in the plane.

Contribution

It determines the entire homomorphism poset of K_{3,3}, providing a detailed classification of geometric realizations and their relationships.

Findings

The homomorphism poset of K_{3,3} is fully characterized.

A classification of geometric realizations of K_{3,3} is provided.

The structure of the partial order induced by geometric homomorphisms is described.

Abstract

A geometric graph \G is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call \G a geometric realization of the underlying abstract graph G. A geometric homomorphism from \G to \H is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). Geometric homomorphisms can be used to define a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G. In this paper, the homomorphism poset of K_{3,3} is determined.