Generalized One-to-One Mappings between Homomorphism Sets of Digraphs

Structural properties of finite digraphs $R$ and $S$ are studied which enforce $\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for every finite digraph $G \in \mathfrak{ D }'$, where ${\cal H}(G,H)$ is the set of homomorphisms from $G$ to $H$, and $\mathfrak{ D }'$ is a class of digraphs. In a previous study, we have seen that the key for such a relation between $R$ and $S$ is the existence of a strong S-scheme from $R$ to $S$. Such an S-scheme $\rho$ defines a one-to-one mapping $\rho_G : {\cal S}(G,R) \rightarrow {\cal S}(G,S)$ for every $G \in \mathfrak{ D }'$, where ${\cal S}(G,H)$ is the set of homomorphisms from $G$ to $H$ mapping proper arcs of $G$ to proper arcs of $H$. In the present article, we characterize S-schemes $\rho$ which are induced by strict homomorphisms $\epsilon : {\cal E}(R) \rightarrow {\cal E}(S)$ between auxiliary systems of $R$ and $S$, and we analyze the mutual dependency between the properties of $\rho$ and $\epsilon$. Wide applicability of the theory is ensured by specifying the auxiliary systems ${\cal E}(R)$ and ${\cal E}(S)$ as EV-systems of $R$ and $S$. The results are applied on a rearrangement method for digraphs and on undirected graphs.