Extremal digraphs for open neighbourhood location-domination and identifying codes

TL;DR

This paper characterizes extremal digraphs where the entire vertex set is needed for open neighbourhood locating-dominating sets, extending previous work on identifying codes and providing structural insights and recursive constructions.

Contribution

It introduces a comprehensive structural analysis and construction methods for extremal digraphs related to open neighbourhood locating-dominating sets and identifying codes.

Findings

Characterization of extremal digraphs with $oldsymbol{oldsymbol{oldsymbol{ ext{}}}}$-set equal to the vertex set

Structural properties of extremal digraphs and their construction methods

Recursive description of extremal di-trees

Abstract

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).