  In this paper we study the problem of testing graph isomorphism (GI) in the
CONGEST distributed model. In this setting we test whether the distributive
network, $G_U$, is isomorphic to $G_K$ which is given as an input to all the
nodes in the network, or alternatively, only to a single node.
  We first consider the decision variant of the problem in which the algorithm
distinguishes $G_U$ and $G_K$ which are isomorphic from $G_U$ and $G_K$ which
are not isomorphic. We provide a randomized algorithm with $O(n)$ rounds for
the setting in which $G_K$ is given only to a single node. We prove that for
this setting the number of rounds of any deterministic algorithm is
$\tilde{\Omega}(n^2)$ rounds, where $n$ denotes the number of nodes, which
implies a separation between the randomized and the deterministic complexities
of deciding GI.
  We then consider the \emph{property testing} variant of the problem, where
the algorithm is only required to distinguish the case that $G_U$ and $G_K$ are
isomorphic from the case that $G_U$ and $G_K$ are \emph{far} from being
isomorphic (according to some predetermined distance measure). We show that
every algorithm requires $\Omega(D)$ rounds, where $D$ denotes the diameter of
the network. This lower bound holds even if all the nodes are given $G_K$ as an
input, and even if the message size is unbounded. We provide a randomized
algorithm with an almost matching round complexity of $O(D+(\epsilon^{-1}\log
n)^2)$ rounds that is suitable for dense graphs.
  We also show that with the same number of rounds it is possible that each
node outputs its mapping according to a bijection which is an
\emph{approximated} isomorphism.
  We conclude with simple simulation arguments that allow us to obtain
essentially tight algorithms with round complexity $\tilde{O}(D)$ for special
families of sparse graphs.
