Logarithmic Weisfeiler–Leman Identifies All Graphs of Bounded Rank Width
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In this paper, we extend the work of Grohe Neuen (ACM T. Comput. Log., 2023) to show that the (6k+3)-dimensional Weisfeiler–Leman (WL) algorithm can identify graphs of rank width k using only O(log n) rounds. As a consequence, we obtain that graphs of bounded rank width are identified by + formulas with 6k+4 variables and quantifier depth O(log n). Furthermore, in light of the parallel WL implementation due to Grohe Verbitsky (ICALP 2006), we obtain ^1 upper bounds for isomorphism testing of graphs of bounded rank width. Prior to this paper, isomorphism testing for graphs of bounded rank width was not known to be in .
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