
Approximations of Isomorphism and Logics with LinearAlgebraic Operators
Invertible map equivalences are approximations of graph isomorphism that...
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On WeisfeilerLeman Invariance: Subgraph Counts and Related Graph Properties
We investigate graph properties and parameters that are invariant under ...
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Counting Bounded Tree Depth Homomorphisms
We prove that graphs G, G' satisfy the same sentences of firstorder log...
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On Cayley graphs of algebraic structures
We present simple graphtheoretic characterizations of Cayley graphs for...
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On the WeisfeilerLeman Dimension of Fractional Packing
The kdimensional WeisfeilerLeman procedure (kWL), which colors ktupl...
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The complexity of generalvalued CSPs seen from the other side
Generalvalued constraint satisfaction problems (VCSPs) are generalisati...
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VC density of set systems defnable in treelike graphs
We study set systems definable in graphs using variants of logic with di...
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On the Expressive Power of Homomorphism Counts
A classical result by Lovász asserts that two graphs G and H are isomorphic if and only if they have the same lefthomomorphism vector, that is, for every graph F, the number of homomorphisms from F to G coincides with the number of homomorphisms from F to H. Dell, Grohe, and Rattan showed that restrictions of the lefthomomorphism vector to a class of graphs can capture several different relaxations of isomorphism, including cospectrality (i.e., two graphs having the same characteristic polynomial), fractional isomorphism and, more broadly, equivalence in counting logics with a fixed number of variables. On the other side, a result by Chaudhuri and Vardi asserts that isomorphism is also captured by the righthomomorphism vector, that is, two graphs G and H are isomorphic if and only if for every graph F, the number of homomorphisms from G to F coincides with the number of homomorphisms from H to F. In this paper, we embark on a study of the restrictions of the righthomomorphism vector by investigating relaxations of isomorphism that can or cannot be captured by restricting the righthomomorphism vector to a fixed class of graphs. Our results unveil striking differences between the expressive power of the lefthomomorphism vector and the righthomomorphism vector. We show that cospectrality, fractional isomorphism, and equivalence in counting logics with a fixed number of variables cannot be captured by restricting the righthomomorphism vector to a class of graphs. In the opposite direction, we show that chromatic equivalence cannot be captured by restricting the lefthomomorphism vector to a class of graphs, while, clearly, it can be captured by restricting the righthomomorphism vector to the class of all cliques.
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