Nordhaus–Gaddum bounds for locating domination

A dominating set S of graph G is called metric-locating–dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S . If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S , then it is said to be locating–dominating. Locating, metric-locating–dominating and locating–dominating sets of minimum cardinality are called β -codes, η -codes and λ -codes, respectively. A Nordhaus–Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its complement G ¯ . In this paper, we present some Nordhaus–Gaddum bounds for the location number β , the metric-location–domination number η and the location–domination number λ . Moreover, in each case, the graph family attaining the corresponding bound is fully characterized.