Logarithmic Weisfeiler–Leman and Treewidth

03/14/2023

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by   Michael Levet, et al.

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In this paper, we show that the (3k+4)-dimensional Weisfeiler–Leman algorithm can identify graphs of treewidth k in O(log n) rounds. This improves the result of Grohe Verbitsky (ICALP 2006), who previously established the analogous result for (4k+3)-dimensional Weisfeiler–Leman. In light of the equivalence between Weisfeiler–Leman and the logic + (Cai, Fürer, Immerman, Combinatorica 1992), we obtain an improvement in the descriptive complexity for graphs of treewidth k. Precisely, if G is a graph of treewidth k, then there exists a (3k+5)-variable formula φ in + with quantifier depth O(log n) that identifies G up to isomorphism.