
WeisfeilerLeman meets Homomorphisms
In this paper, we relate a beautiful theory by Lovász with a popular heu...
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Colorcritical Graphs and Hereditary Hypergraphs
A quick proof of Gallai's celebrated theorem on colorcritical graphs is...
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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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Identifiability of Graphs with Small Color Classes by the WeisfeilerLeman Algorithm
As it is well known, the isomorphism problem for vertexcolored graphs w...
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Comparative DesignChoice Analysis of Color Refinement Algorithms Beyond the Worst Case
Color refinement is a crucial subroutine in symmetry detection in theory...
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Walk refinement, walk logic, and the iteration number of the WeisfeilerLeman algorithm
We show that the 2dimensional WeisfeilerLeman algorithm stabilizes nv...
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The Iteration Number of Colour Refinement
The Colour Refinement procedure and its generalisation to higher dimensi...
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Color Refinement, Homomorphisms, and Hypergraphs
Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the wellknown colorrefinement algorithm does not distinguish two graphs G and H if and only if, for every tree T, the number of homomorphisms Hom(T,G) from T to G is equal to the corresponding number Hom(T,H) from T to H (Dell, Grohe, Rattan 2018). We show how this approach transfers to hypergraphs by introducing a generalization of color refinement. We prove that it does not distinguish two hypergraphs G and H if and only if, for every connected Bergeacyclic hypergraph B, we have Hom(B,G) = Hom(B,H). To this end, we show how homomorphisms of hypergraphs and of a colored variant of their incidence graphs are related to each other. This reduces the above statement to one about vertexcolored graphs.
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