Nordhaus–Gaddum bounds for locating domination

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