Open neighborhood locating–dominating in trees

Abstract

For a graph G that models a facility or a multiprocessor network, detection devices can be placed at the vertices so as to identify the location of an intruder such as a thief or saboteur or a faulty processor. Open neighborhood locating–dominating sets are of interest when the intruder/fault at a vertex precludes its detection at that location. The parameter OLD ( G ) denotes the minimum cardinality of a vertex set S ⊆ V ( G ) such that for each vertex v in V ( G ) its open neighborhood N ( v ) has a unique non-empty intersection with S . For a tree T n of order n we have ⌈ n / 2 ⌉ + 1 ≤ OLD ( T n ) ≤ n − 1 . We characterize the trees that achieve these extremal values.