Abstract
A function f of a graph is called a complete graph invariant if two given graphs G and H are isomorphic exactly when f(G) = f(H). If additionally, f(G) is a graph isomorphic to G, then f is called a canonical form for graphs. Gurevich [9] proves that any polynomial-time computable complete invariant can be transformed into a polynomial-time computable canonical form. We extend this equivalence to the polylogarithmic-time model of parallel computation for classes of graphs having either bounded rigidity index or small separators. In particular, our results apply to three representative classes of graphs embeddable into a fixed surface, namely, to 3-connected graphs admitting either a polyhedral or a large-edge-width embedding as well as to all embeddable 5-connected graphs. Another application covers graphs with treewidth bounded by a constant k. Since for the latter class of graphs a complete invariant is computable in NC, it follows that graphs of bounded treewidth have a canonical form (and even a canonical labeling) computable in NC.
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Köbler, J., Verbitsky, O. (2008). From Invariants to Canonization in Parallel. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_23
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DOI: https://doi.org/10.1007/978-3-540-79709-8_23
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