A set $S$ of vertices of a digraph $D$ is called an open neighbourhood locating-dominating set if every vertex in $D$ has an in-neighbour in $S$, and for every pair $u,v$ of vertices of $D$, there is a vertex in $S$ that is an in-neighbour of exactly one of $u$ and $v$. The smallest size of an open neighbourhood locating-dominating set of a digraph $D$ is denoted by $\gamma_{OL}(D)$. We study the class of digraphs $D$ whose only open neighbourhood locating-dominating set consists of the whole set of vertices, in other words, $\gamma_{OL}(D)$ is equal to the order of $D$. We call those digraphs extremal. By considering digraphs with loops allowed, our definition also applies to the related (and more widely studied) concept of identifying codes. We extend previous studies from the literature for both open neighbourhood locating-dominating sets and identifying codes of both undirected and directed graphs. These results all correspond to studying open neighbourhood locating-dominating sets on special classes of digraphs. To do so, we prove general structural properties of extremal digraphs, and we describe how they can all be constructed. We then use these properties to give new proofs of several known results from the literature. We also give a recursive and constructive characterization of the extremal di-trees (digraphs whose underlying undirected graph is a tree).

翻译：设$D$为有向图，其顶点子集$S$称为开邻域定位-支配集，若$D$中每个顶点在$S$中均有一个入邻点，且对于$D$中任意一对顶点$u,v$，存在$S$中一个顶点恰好是$u$或$v$中一者的入邻点。有向图$D$的最小开邻域定位-支配集的大小记为$\gamma_{OL}(D)$。本文研究一类仅以全体顶点集为开邻域定位-支配集的有向图$D$，即$\gamma_{OL}(D)$等于$D$的阶。我们称这类有向图为极值图。通过允许有向图中存在环，该定义亦适用于相关且更广泛研究的标识码概念。我们扩展了已有文献中对无向图与有向图的开邻域定位-支配集及标识码的研究，这些结果均对应于特殊有向图类上的开邻域定位-支配集问题。为此，我们证明了极值有向图的一般结构性质，并描述了其构造方法。利用这些性质，我们给出了文献中若干已知结论的新证明。此外，我们给出了极值有向树（其底图无向图为树的有向图）的递归构造刻画。