Extremal digraphs for open neighbourhood location-domination and identifying codes

TL;DR

This work studies extremal digraphs for open neighbourhood locating-dominating (OLD) sets, i.e., digraphs D with γ_{OL}(D)=|V(D)|. It develops structural tools around forced vertices and forcing arcs, showing that the forcing-arcs subgraph forms a spanning disjoint union of directed cycles and that the remaining structure is captured by the acyclic digraph H(D). The authors provide a complete constructive characterization of extremal digraphs and a recursive, explicit description for extremal di-trees, along with new proofs of key existing results in the literature. The results unify OLD sets and identifying codes in digraphs, extend known characterizations, and offer a constructive blueprint for building all extremal objects, with implications for related combinatorial problems and potential algorithmic applications.

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 $γ_{OL}(D)$. We study the class of digraphs $D$ whose only open neighbourhood locating-dominating set consists of the whole set of vertices, in other words, $γ_{OL}(D)$ 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).

Figures (4)

• Figure 1: The four connected locatable digraphs of order 1 and 2. Forcing arcs are dashed.
• Figure 2: An example of the construction from Theorem \ref{['thm:construct']}. The square vertex is the only one in $V_d$; the circled vertices are those in $V_l$; the dashed arcs are the forcing arcs; the wriggled arcs are those of $\mathcal{H}(D)$.
• Figure 3: Two extremal di-trees of order 3. Forcing arcs are dashed.
• Figure 4: All extremal di-trees with order at most 4. Forcing arcs are dashed.

Theorems & Definitions (52)

• Theorem 1: GM07
• Theorem 2: FNP13
• Theorem 3: FGRS21
• Definition 4
• Proposition 5
• proof
• Corollary 6
• Theorem 7: Bondy's Theorem B72
• Corollary 8
• proof