  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.
