Extremal digraphs for open neighbourhood location-domination and identifying codes

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.