Abstract
Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 26 March 2020Accepted: 11 May 2021Published online: 19 August 2021Keywordsgraph isomorphism, Weisfeiler--Leman algorithm, Cai--Fürer--Immerman graphs, coherent configurationsAMS Subject Headings03D15, 68Q25Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec
Highlights
Over 50 years ago, Weisfeiler and Leman [34] described a natural combinatorial procedure that since constantly plays a significant role in the research on the graph isomorphism problem
The problem of deciding whether a given vertex-colored graph with maximum color multiplicity 4 is amenable to 2-dimensional Weisfeiler-Leman algorithm (2-WL) is solvable in P
The Cut-Down Lemma and the preceding analysis in Section 3 reduce our task to deciding separability of a coherent configuration C under the following three conditions: (1) C is indecomposable, (2) all fibers of C have size 4, (3) every non-uniform interspace of C is of type 2K2,2
Summary
Over 50 years ago, Weisfeiler and Leman [34] described a natural combinatorial procedure that since constantly plays a significant role in the research on the graph isomorphism problem. We consider vertex-colored graphs with the color multiplicity, that is, the maximum number of colored vertices, as parameter If this parameter is bounded, the graph isomorphism problem is known to be efficiently solvable. The problem of deciding whether a given vertex-colored graph with maximum color multiplicity 4 is amenable to 2-WL is solvable in P. This holds true for vertexand edge-colored directed graphs. The proof of Theorem 1.2 yields an algorithm deciding amenability of graphs of color multiplicity at most 4 with running time O(n2+ω), where ω < 2.373 is the exponent of fast matrix multiplication [19]. All proofs omitted in this version of the paper can be found in [17]
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