  In [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action
of local Clifford operations on graph states in terms of a graph transformation
rule, known in graph theory as \emph{local complementation}. It was shown that
two graph states are equivalent under the local Clifford group if and only if
there exists a sequence of local complementations which relates their
associated graphs. In this short note we report the existence of a polynomial
time algorithm, published in [Combinatorica 11 (4), 315 (1991)], which decides
whether two given graphs are related by a sequence of local complementations.
Hence an efficient algorithm to detect local Clifford equivalence of graph
states is obtained.
