2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128605It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties of regular $k$-partitions that, on one hand, generalize automorphic $k$-partitions (=systems of $k$-orbits of permutation groups), and, on other hand, schemes of relations (strongly regular 2-partitions or regular 3-partitions), that are a subject of the algebraic combinatorics. It is shown that the stabilization of a graph by quadrangles detects the triviality of the graph automorphism group. The result is obtained by lineariation of the algebraic combinatorics. Keywords: $k$-partitions, symmetry, algebraic combinatorics9 pages, Latex2eGeneral Mathematics05C60, 05E99text