The Weisfeiler-Leman algorithm

Abstract

More than half a century after its first formulation, the Weisfeiler-Leman (WL) algorithm is still an important combinatorial technique whenever graphs or other relational structures are to be classified. However, despite its simple algebraic description and its variety of applications, we still lack a precise understanding of the expressive power of the algorithm. This column introduces the reader to the basic concepts of the WL algorithm and discusses its dimension as a parameter to capture the structural complexity of an input graph. Specifically, I present a survey of work regarding the WL dimension conducted with my co-authors. First, I outline the proof that the 3-dimensional WL algorithm (3-WL) is able to identify every planar graph. The proof version presented here relies on strong insights about the ability of 2-WL to decompose graphs. Afterwards, I highlight the most important ingredients of the generalisation of our bound to graphs that are parameterised by their Euler genus. Further details as well as a study of other aspects of the WL algorithm can be found in my dissertation [Kiefer 2020].