Testing Isomorphism of Graphs in Polynomial Time
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Given a graph G, the graph [G] obtained by adding, for each pair of vertices of G, a unique vertex adjacent to both vertices is called the binding graph of G. In this work, we show that the class of binding graphs is graph-isomorphism complete and that the stable partitions of binding graphs by the Weisfeiler-Lehman (WL) algorithm produce automorphism partitions. To test the isomorphism of two graphs G and H, one computes the stable graph of the binding graph [G⊎ H] for the disjoint union graph G⊎ H. The automorphism partition reveals the isomorphism of G and H. Because the WL algorithm is a polynomial-time procedure, the claim can be made that the graph-isomorphism problem is in complexity class 𝙿.
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