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