  A graph $G$ is $H$-saturated for a graph $H$, if $G$ does not contain a copy
of $H$ but adding any new edge to $G$ results in such a copy. An $H$-saturated
graph on a given number of vertices always exists and the properties of such
graphs, for example their highest density, have been studied intensively. A
graph $G$ is $H$-induced-saturated if $G$ does not have an induced subgraph
isomorphic to $H$, but adding an edge to $G$ from its complement or deleting an
edge from $G$ results in an induced copy of $H$. It is not immediate anymore
that $H$-induced-saturated graphs exist. In fact, Martin and Smith (2012)
showed that there is no $P_4$-induced-saturated graph. Behrens et.al. (2016)
proved that if $H$ belongs to a few simple classes of graphs such as a class of
odd cycles of length at least $5$, stars of size at least $2$, or matchings of
size at least $2$, then there is an $H$-induced-saturated graph. This paper
addresses the existence question for $H$-induced-saturated graphs. It is shown
that Cartesian products of cliques are $H$-induced-saturated graphs for $H$ in
several infinite families, including large families of trees. A complete
characterization of all connected graphs $H$ for which a Cartesian product of
two cliques is an $H$-induced-saturated graph is given. Finally, several
results on induced saturation for prime graphs and families of graphs are
provided.
