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.
Categories: math.CO