  Let $v$ be a vertex of a graph $G$. By the local complementation of $G$ at
$v$ we mean to complement the subgraph induced by the neighbors of $v$. This
operator can be generalized as follows. Assume that, each edge of $G$ has a
label in the finite field $\mathbf{F}_q$. Let $(g_{ij})$ be set of labels
($g_{ij}$ is the label of edge $ij$). We define two types of operators. For the
first one, let $v$ be a vertex of $G$ and $a\in \mathbf{F}_q$, and obtain the
graph with labels $g'_{ij}=g_{ij}+ag_{vi}g_{vj}$. For the second, if $0\neq
b\in \mathbf{F}_q$ the resulted graph is a graph with labels $g''_{vi}=bg_{vi}$
and $g''_{ij}=g_{ij}$, for $i,j$ unequal to $v$. It is clear that if the field
is binary, the operators are just local complementations that we described.
  The problem of whether two graphs are equivalent under local complementations
has been studied, \cite{bouchalg}. Here we consider the general case and
assuming that $q$ is odd, present the first known efficient algorithm to verify
whether two graphs are locally equivalent or not.
