  Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set
$V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed
\emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$,
and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for
short) if no two vertices in $G$ have the same set of neighbors in $S$, and
each vertex in $S$ is open-dominated exactly once by $S$. The problem of
deciding whether or not $G$ has an $OLD_{oind}$-set has important applications
that have been reported elsewhere. As the problem is known to be
$\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that
its complexity remains the same even for just planar bipartite graphs of
maximum degree five and girth six, and also for planar subcubic graphs of girth
nine. Also, we present characterizations of both $P_4$-tidy graphs and the
complementary prisms of cographs that have an $OLD_{oind}$-set.

Discrete Mathematics