Locating domination in bipartite graphs and their complements

A set $S$ of vertices of a graph $G$ is \emph{distinguishing} if the sets of neighbors in $S$ for every pair of vertices not in $S$ are distinct. A \emph{locating-dominating set} of $G$ is a dominating distinguishing set. The \emph{location-domination number} of $G$, $\lambda(G)$, is the minimum cardinality of a locating-dominating set. In this work we study relationships between $\lambda({G})$ and $\lambda (\overline{G})$ for bipartite graphs. The main result is the characterization of all connected bipartite graphs $G$ satisfying $\lambda (\overline{G})=\lambda({G})+1$. To this aim, we define an edge-labeled graph $G^S$ associated with a distinguishing set $S$ that turns out to be very helpful.