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On a Theorem of Lovász that hom(⋅, H) Determines the Isomorphism Type of H

Authors Jin-Yi Cai, Artem Govorov

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Jin-Yi Cai
  • Department of Computer Sciences, University of Wisconsin-Madison, USA
Artem Govorov
  • Department of Computer Sciences, University of Wisconsin-Madison, USA


We thank the anonymous referees for helpful comments and suggestions. We also thank Guus Regts for telling us of his work, and for very insightful comments on this paper.

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Jin-Yi Cai and Artem Govorov. On a Theorem of Lovász that hom(⋅, H) Determines the Isomorphism Type of H. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 17:1-17:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


Graph homomorphism has been an important research topic since its introduction [László Lovász, 1967]. Stated in the language of binary relational structures in that paper [László Lovász, 1967], Lovász proved a fundamental theorem that the graph homomorphism function G ↦ 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 Lovász and others [László Lovász, 2006; Michael Freedman et al., 2007; Christian Borgs et al., 2008; Alexander Schrijver, 2009; László Lovász and Balázs Szegedy, 2009]. 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 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, i.e., it provides an algorithm that for any H either outputs a P-time algorithm solving hom(⋅, H) or a P-time reduction from a canonical #P-hard problem to hom(⋅, H).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Problems, reductions and completeness
  • Graph homomorphism
  • Partition function
  • Complexity dichotomy
  • Connection matrices and tensors


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