Given graphs X and Y, we define two feasibility programs which we show have a solution over the completely positive cone if and only if there exists a homomorphism from X to Y. By varying the cone, we obtain similar characterizations of quantum/entanglement-assisted and three previously studied relaxations of these relations. Motivated by this, we investigate the properties of these conic homomorphisms for general (suitable) cones. We also consider two generalized versions of the Lovasz theta function, and how they interact with these homomorphisms. We prove analogs of several results on classical graph as well as some monotonicity theorems. We also show that one of the generalized theta functions is multiplicative on lexicographic and disjunctive graph products.