
The Iteration Number of Colour Refinement
The Colour Refinement procedure and its generalisation to higher dimensi...
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Logarithmic WeisfeilerLeman Identifies All Planar Graphs
The WeisfeilerLeman (WL) algorithm is a wellknown combinatorial proced...
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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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Color Refinement, Homomorphisms, and Hypergraphs
Recent results show that the structural similarity of graphs can be char...
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On the signed chromatic number of some classes of graphs
A signed graph (G, σ) is a graph G along with a function σ: E(G) →{+,}....
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WeisfeilerLeman meets Homomorphisms
In this paper, we relate a beautiful theory by Lovász with a popular heu...
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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 nvertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3variable first order logic with counting, then they can also be distinguished in this logic by a formula of quantifier depth at most O(n log n). For this we exploit a new refinement based on counting walks and argue that its iteration number differs from the classic WeisfeilerLeman refinement by at most a logarithmic factor. We then prove matching linear upper and lower bounds on the number of iterations of the walk refinement. This is achieved with an algebraic approach by exploiting properties of semisimple matrix algebras. We also define a walk logic and a bijective walk pebble game that precisely correspond to the new walk refinement.
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