Interdependencies of less-equal-relations between partial Lov\'{a}sz-vectors of digraphs

Abstract

For digraphs $G$ and $H$, let ${\cal H}(G,H)$ be the set of all homomorphisms from $G$ to $H$, and let ${\cal S}(G,H)$ be the subset of those homomorphisms mapping all proper arcs in $G$ to proper arcs in $H$. From an earlier investigation we know that for certain digraphs $R$ and $S$, the relation "$\# {\cal S}(G,R) \leq \# {\cal S}(G,S)$ for all $G \in \mathfrak{ D }'$" implies "$\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for all $G \in \mathfrak{ D }'$", where $\mathfrak{ D }'$ is a subclass of digraphs. Now we ask for the inverse: For which digraphs $R, S$ and which subclasses $\mathfrak{ D }'$ of digraphs does "$\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for all $G \in \mathfrak{ D }'$" imply "$\# {\cal S}(G,R) \leq \# {\cal S}(G,S)$ for all $G \in \mathfrak{ D }'$"? We prove this implication for three combinations of digraph classes. In particular, the relations are equivalent for all flat posets $R, S$ with respect to all flat posets $G$.