On a Theorem of Lov\'asz that $\hom(\cdot, H)$ Determines the Isomorhphism Type of $H$

Graph homomorphism has been an important research topic since its introduction [13]. Stated in the language of binary relational structures in this paper [13], Lov\'asz proved a fundamental theorem that the graph homomorphism function $G \mapsto \hom(G, H)$ for 0-1 valued $H$ (as the adjacency matrix of a graph) determines the isomorphism type of $H$. In the past 50 years various extensions have been proved by Lov\'asz and others [14, 9, 1, 18, 16]. These extend the basic 0-1 case to admit vertex and edge weights; but always with some restrictions such as all vertex weights must be positive. In this paper we prove a general form of this theorem where $H$ can have arbitrary vertex and edge weights. An innovative aspect is we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective, i.e., an algorithm such that for any $H$ either outputs a P-time algorithm solving $\hom(\cdot, H)$ or a P-time reduction from a canonical #P-hard problem to $\hom(\cdot, H)$.