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 $D$, which we call \emph{extremal}. By considering digraphs with loops allowed, our definition also applies to the related (and more widely studied) concept of identifying codes. Extending some previous studies from the literature for both open neighbourhood locating-dominating sets and identifying codes of both undirected and directed graphs (which all correspond to studying special classes of digraphs), we prove general structural properties of such 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 digraphs whose underlying undirected graph is a tree.

翻译：设$D$为有向图，若其顶点子集$S$满足：$D$中每个顶点在$S$中均有一个入邻点，且对$D$中任意两个顶点$u,v$，存在$S$中一个顶点恰好是$u$或$v$之一的入邻点，则称$S$为开邻域定位支配集。记$\gamma_{OL}(D)$为有向图$D$的最小开邻域定位支配集的大小。本文研究仅以全体顶点作为开邻域定位支配集的有向图类，即$\gamma_{OL}(D)$等于$D$的阶数，我们称之为极值有向图。通过允许有向图包含自环，我们的定义同样适用于相关（且被广泛研究）的标识码概念。延续此前文献中对无向图与有向图（均对应于特殊有向图类）的开邻域定位支配集和标识码的部分研究，我们证明了此类极值有向图的一般结构性质，并描述了其构造方法。进而利用这些性质给出了文献中若干已知结果的新证明。此外，我们给出了底图为树的极值有向图的一种递归构造性刻画。