  It 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 combinatorics
