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 $\frac{2}{3}\left(n -γ^{LD}(G)\right)$, 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 $\frac{2}{5}$. 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.

翻译：自1980年代Slater和Rall引入定位支配码以来，该课题得到了广泛研究。本文聚焦于图中必须属于所有最小定位支配码的顶点，我们称其为"最小强制顶点"。研究表明，在阶数为$n$的连通非平凡图中，最小强制顶点数上界为$\frac{2}{3}\left(n -γ^{LD}(G)\right)$，其中$γ^{LD}(G)$表示最小定位支配码的基数。由此推出连通非平凡图中最小强制顶点数与阶数的最大比值不超过$\frac{2}{5}$。这两个界均可达到，特别地，比值$\frac{2}{5}$由具有唯一最小定位支配码（大小为$2m$）的$5m$阶路径图实现。作为自然推广，我们还确定了所有阶数路径图中不同最小定位支配码的数量。此外，证明判定顶点是否为最小强制顶点属于co-NP难问题。