Homomorphic preimages of geometric paths.

A graph G is a homomorphic preimage of another graph H, or equivalently G is H-colorable, if there exists a graph homomorphism f : G -> H. A geometric graph (G) over bar is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. A geometric homomorphism (respectively, isomorphism) (G) over bar -> (H) over bar is a graph homomorphism (respectively, isomorphism) that preserves edge crossings (respectively, and non-crossings). The homomorphism poset G of a graph G is the set of isomorphism classes of geometric realizations of G partially ordered by the existence of injective geometric homomorphisms. A geometric graph (G) over bar is H-colorable if (G) over bar -> (H) over bar for some (H) over bar -> H. In this paper, we provide necessary and sufficient conditions for (G) over bar to be P-n-colorable for n >= 2. Along the way, we also provide necessary and sufficient conditions for (G) over bar to be K-2,K-3-colorable.