An efficient algorithm to recognize locally equivalent graphs

There are local operators on (labeled) graphs G with labels (gij) in a finite field. If the field is binary, the operations are just the local complementations, but in the general case, there are two different types of operators. For the first type, let v be a vertex of the graph and a ∈ Fq, and we obtain a graph with labels g ′ ij = gij+agvigvj . For the second type of operators, let 0 6= b ∈ Fq and the resulted graph is a graph with labels g vi = bgvi and g ′′ ij = gij , for i, j unequal to v. We present the first known efficient algorithm to verify whether two graphs are locally equivalent or not.