  In this paper we present distributed testing algorithms of graph properties
in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error
testing algorithms in the general graph model.
  We first describe a general procedure for converting $\epsilon$-testers with
a number of rounds $f(D)$, where $D$ denotes the diameter of the graph, to
$O((\log n)/\epsilon)+f((\log n)/\epsilon)$ rounds, where $n$ is the number of
processors of the network. We then apply this procedure to obtain an optimal
tester, in terms of $n$, for testing bipartiteness, whose round complexity is
$O(\epsilon^{-1}\log n)$, which improves over the $poly(\epsilon^{-1} \log
n)$-round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for
cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects
the graph so that the corrected graph is acyclic. Note that, unlike a tester, a
corrector needs to mend the graph in many places in the case that the graph is
far from having the property.
  In the second part of the paper we design algorithms for testing whether the
network is $H$-free for any connected $H$ of size up to four with round
complexity of $O(\epsilon^{-1})$. This improves over the
$O(\epsilon^{-2})$-round algorithms for testing triangle freeness by
Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size $4$ by
Fraigniaud et al. (DISC 2016).
  In the last part we generalize the global tester by Iwama and Yoshida (ITCS
2014) of testing $k$-path freeness to testing the exclusion of any tree of
order $k$. We then show how to simulate this algorithm in the CONGEST-model in
$O(k^{k^2+1}\cdot\epsilon^{-k})$ rounds.
