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 \emph{min-forced vertices}. We show that the number of min-forced vertices in a connected nontrivial graph of order $n$ is bounded above by $\frac{2}{3}\left(n -\gamma^{LD}(G)\right)$, where $\gamma^{LD}(G)$ 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 $\frac{2}{5}$. Moreover, both of these bounds can be attained. In particular, the ratio $\frac{2}{5}$ 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.