New Results on Vertices that Belong to Every Minimum Locating-Dominating Code

Locating-dominating codes have been studied widely since their introduction in the 1980s by Slater and Rall. In this paper, we concentrate on vertices that must belong to all minimum locating-dominating codes in a graph. We call them min-forced vertices. We show that the number of min-forced vertices in a connected nontrivial graph of order n is bounded above by 2 3 n − γ LD(G)  , where γ LD(G) denotes the cardinality of a minimum locating-dominating code. This implies that the maximum ratio between the number of min-forced vertices and the order of a connected nontrivial graph is at most 2 5 . Moreover, both of these bounds can be attained. In particular, the ratio 2 5 is obtained by paths of order 5m having a unique minimum locating-dominating code of size 2m. Furthermore, as a natural extension, we determine the number of different minimum locating-dominating codes in paths of all orders. In addition, we show that deciding whether a vertex is min-forced is co-NP-hard.