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arXiv · 1901.10330

Canonisation and Definability for Graphs of Bounded Rank Width

Martin Grohe·Daniel Neuen
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Abstract

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width $k$ is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time $n^{f(k)}$ for a non-elementary function $f$. Our result yields an isomorphism test for graphs of rank width $k$ running in time $n^{O(k)}$. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.

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Martin Grohe, Daniel Neuen. 2019-01-29. Canonisation and Definability for Graphs of Bounded Rank Width. https://doi.org/10.1145/3568025

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