  The distributed subgraph detection asks, for a fixed graph $H$, whether the
$n$-node input graph contains $H$ as a subgraph or not. In the standard CONGEST
model of distributed computing, the complexity of clique/cycle detection and
listing has received a lot of attention recently.
  In this paper we consider the induced variant of subgraph detection, where
the goal is to decide whether the $n$-node input graph contains $H$ as an
\emph{induced} subgraph or not. We first show a $\tilde{\Omega}(n)$ lower bound
for detecting the existence of an induced $k$-cycle for any $k\geq 4$ in the
CONGEST model. This lower bound is tight for $k=4$, and shows that the induced
variant of $k$-cycle detection is much harder than the non-induced version.
This lower bound is proved via a reduction from two-party communication
complexity. We complement this result by showing that for $5\leq k\leq 7$, this
$\tilde{\Omega}(n)$ lower bound cannot be improved via the two-party
communication framework.
  We then show how to prove stronger lower bounds for larger values of $k$.
More precisely, we show that detecting an induced $k$-cycle for any $k\geq 8$
requires $\tilde{\Omega}(n^{2-\Theta{(1/k)}})$ rounds in the CONGEST model,
nearly matching the known upper bound $\tilde{O}(n^{2-\Theta{(1/k)}})$ of the
general $k$-node subgraph detection (which also applies to the induced version)
by Eden, Fiat, Fischer, Kuhn, and Oshman~[DISC 2019].
  Finally, we investigate the case where $H$ is the diamond (the diamond is
obtained by adding an edge to a 4-cycle, or equivalently removing an edge from
a 4-clique), and show non-trivial upper and lower bounds on the complexity of
the induced version of diamond detecting and listing.
