The Mutual-Visibility Problem In Directed Graphs

TL;DR

This work extends the mutual-visibility concept from undirected graphs to directed graphs, introducing $\mu(D)$ and related notions, and characterizing their behavior across key graph classes. It proves $\mu(D)=1$ for directed acyclic graphs and $\mu(C_n)=2$ for directed cycles ($n\ge 3$), while showing that tournaments can host arbitrarily large mutual-visibility sets, with Paley tournaments providing constructive lower bounds that scale with tournament size. The paper also provides a polynomial-time verification procedure for candidate sets but proves that computing $\mu(D)$ is NP-hard for general digraphs via a reduction from the undirected case, and it analyzes structural bounds via strongly connected components and strong bridges. These results establish a foundational understanding of directed mutual visibility and motivate future work on variants and condensation-graph hierarchies with potential applications in directed networks.

Abstract

The mutual-visibility problem, originally defined for undirected graphs, asks for the size of the maximum set of vertices $S$ such that every pair of vertices in $S$ is connected by a shortest path passing only through vertices in $V \setminus S$. In this paper, we extend this concept to directed graphs, establishing fundamental results for several graph classes. We prove that for Directed Acyclic Graphs (DAGs), the mutual-visibility number $μ(D)$ is always 1, and for directed cycles of length $n \geq 3$, it is strictly 2. In contrast, we demonstrate that tournaments can support arbitrarily large mutual-visibility sets; specifically, using properties of Paley tournaments, we show that $μ(T)$ grows linearly with the size of the tournament. On the algorithmic side, we show that while verifying a candidate set is polynomial-time solvable ($O(|S|(|V|+|A|))$), the problem of determining $μ(D)$ is NP-hard for general digraphs. We also analyze the impact of strong bridges and strongly connected components on the upper bounds of $μ(D)$.

Figures (1)

• Figure 1: Directed cycle with 8 vertices: $x$ and $y$ both direct to $z$, with remaining vertices forming a bidirectional path from $x$ to $y$. And, an undirected 8-cycle variant.

Theorems & Definitions (21)

• definition 1
• definition 2: Mutual-visibility set
• remark 1
• definition 3: Total mutual-visibility set
• definition 4: Outer mutual-visibility set
• definition 5: Dual mutual-visibility set
• Proposition 2.1
• definition 6: Condensation graph
• remark 2
• Theorem 2.2