Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a Theorem of Lov\'asz that $\hom(\cdot, H)$ Determines the
  Isomorphism Type of $H$   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 $\hom(\cdot, H)$.

24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an