Computing the maximal canonical form for trees in polynomial time

Abstract

Known algorithms computing a canonical form for trees in linear time use specialized canonical forms for trees and no canonical forms defined for all graphs. For a graph $$G=(V,E)$$ the maximal canonical form is obtained by relabelling the vertices with $$1,\ldots ,|V|$$ in a way that the binary number with $$|V|^2$$ bits that is the result of concatenating the rows of the adjacency matrix is maximal. This maximal canonical form is not only defined for all graphs but even plays a special role among the canonical forms for graphs due to some nesting properties allowing orderly algorithms. We give an $$O(|V|^2)$$ algorithm to compute the maximal canonical form of a tree.