Nordhaus-Gaddum bounds for locating domination

Abstract

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 View the MathML source. 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.