On a Theorem of Lovász that $\hom(\cdot, H)$ Determines the Isomorphism Type of $H$

Abstract: Graph homomorphism has been an important research topic since its introduction [17]. Stated in the language of binary relational structures in that paper [17], Lov\'asz proved a fundamental theorem that, for a graph $H$ given by its $0$-$1$ valued adjacency matrix, the graph homomorphism function $G \mapsto \hom(G, H)$ determines the isomorphism type of $H$. In the past 50 years various extensions have been proved by many researchers [18, 12, 1, 23, 21]. These extend the basic $0$-$1$ case to admit vertex and edge weights; but these extensions all have 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. A noteworthy aspect is that 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 in the following sense: it provides an algorithm 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 $\hom(\cdot, H)$.