Weisfeiler-Leman Graph Isomorphism

Color Refinement Algorithm & Graph Distinguishability
Select an example and run WL

1-WL / Color Refinement

c⁽⁰⁾(v) = deg(v)    (or constant)

c⁽ᵗ⁺¹⁾(v) = hash(c⁽ᵗ⁾(v), sort{c⁽ᵗ⁾(u) : u ∈ N(v)})

WL iteratively refines node colors based on neighborhood multisets. Two graphs are potentially isomorphic if their color histograms match at fixpoint. WL is sound but not complete: the Cai-Fürer-Immerman (CFI) graphs (1992) are non-isomorphic yet WL-equivalent. The k-WL hierarchy (Grohe 2017) connects to counting homomorphisms and graph neural networks — 1-WL = message passing GNN expressive power (Xu et al. 2019).