Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

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, has been studied elsewhere and is known to be $\mathcal{NP}$-complete. Reinforcing this result, it appears that the problem is notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs, that have an $OLD_{oind}$-set.