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 \rightarrow H $. A geometric graph $ \overline{G} $ 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) $ \overline{G} \rightarrow \overline{H} $ is a graph homomorphism (respectively, isomorphism) that preserves edge crossings (respectively, and non-crossings). The homomorphism poset \( \mathcal{G} \) 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 $ \overline{G} $ is \( \mathcal{H} \)-colorable if $ \overline{G} \rightarrow \overline{H} $ for some \( \overline{H} \in \mathcal{H} \). In this paper, we provide necessary and sufficient conditions for $ \overline{G} $ to be \( \mathcal{P}_n \)-colorable for $ n \ge 2 $. Along the way, we also provide necessary and sufficient conditions for $ \overline{G} $ to be \( \mathcal{K}_{2,3} \)-colorable.