On Weisfeiler-Leman Invariance: Subgraph Counts and Related Graph Properties

Abstract

The k-dimensional Weisfeiler-Leman algorithm (k-WL) is a fruitful approach to the Graph Isomorphism problem. 2-WL corresponds to the original algorithm suggested by Weisfeiler and Leman over 50 years ago. 1-WL is the classical color refinement routine. Indistinguishability by k-WL is an equivalence relation on graphs that is of fundamental importance for isomorphism testing, descriptive complexity theory, and graph similarity testing which is also of some relevance in artificial intelligence. Focusing on dimensions k = 1, 2, we investigate subgraph patterns whose counts are k-WL invariant, and whose occurrence is k-WL invariant. We achieve a complete description of all such patterns for dimension k = 1 and considerably extend the previous results known for k = 2.

The Iteration Number of Colour Refinement

“…For k > 2, regarding upper bounds on the iteration number of k-WL, without further knowledge about the input graph, no significant improvements over the trivial upper bound n k − 1 are known. 1 Still, when the input graph has bounded treewidth or is a 3-connected planar graph, polylogarithmic upper bounds on the iteration number of k-WL needed to identify the graph are known [17,35].…”

“…Recent results give new upper bounds on the dimension needed for certain interesting graph classes [15,16]. A closely-related direction of research investigates what properties the Weisfeiler-Leman algorithm can detect in graphs [1,8,10].…”

“…We can insert an isolated vertex w into G and insert edges from w to every vertex v ∈ V (G) with deg(v) = d. In the new graph G , the vertex w has a degree other than d + 1, whereas all other vertices have degree d + 1. Thus, the colour classes in π 1 G are {w} and V (G ) \ {w}. After the second iteration, the neighbours of w are distinguished from all other vertices, just like they are in π 1 G .…”

On the construction of cospectral nonisomorphic bipartite graphs

“…In [6], Haemers conjectured that almost all graphs are DS. Recently, in [1], Arvind et al proved that almost all graphs are determined up to isomorphism by their eigenvalues and angles. For surveys on DS and cospectral graphs, we refer to [12,13].…”

Graph Structure and Isomorphism Learning: A Survey

“…Furthermore, enumerating large graphs for higher-order substructure counting is time consuming, hindering the popularity of this approach and its applicability in practice. However, recent advances in discovering theoretical expressive power [85] of substructures and their practical representation capacity of real complex topology reveal their potential to develop more powerful graph representation systems.…”