The homomorphism poset of K2, n

A geometric graph G 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. In this paper, the homomorphism poset of K2,n is determined by establishing a correspondence between realizations of K2,n and permutations of Sn, in which edge crossings correspond to inversions. Through this correspondence, geo-isomorphism defines an equivalence relation on Sn, which we call geo-equivalence. The number of geo-equivalence 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 diagram of the poset structure are given for n ≤ 5.