The Mutual-Visibility Problem In Directed Graphs

The study of mutual visibility has traditionally focused on undirected graphs, asking for the maximum number of vertices that can communicate via shortest paths without intermediate interference from other set members. 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 $n\ge3$, 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 $μ(D)$.

💡 Research Summary

The paper extends the concept of mutual visibility, originally studied in undirected graphs, to the realm of directed graphs (digraphs). After redefining a mutual‑visibility set for digraphs—requiring that for any two distinct vertices x and y in the set there exist both a shortest directed x‑to‑y path and a shortest directed y‑to‑x path whose internal vertices are outside the set—the authors explore structural properties that determine the mutual‑visibility number μ(D).

A central observation is that any mutual‑visibility set must be wholly contained within a single strongly connected component (SCC). The authors prove that if a set S spans more than one SCC, the required bidirectional shortest paths would induce a directed cycle in the condensation graph, contradicting its acyclic nature. Consequently, μ(D) equals the maximum μ(C) over all SCCs C of D. This reduction simplifies the problem: one can treat each SCC independently when searching for the largest mutual‑visibility set.

The paper then investigates the role of strong bridges—edges whose removal increases the number of SCCs. Lemma 2.4 shows that a strong bridge defines a unique source‑sink partition such that every directed path from the source side to the sink side must traverse the bridge. Corollary 2.5 derives a stringent restriction: if a mutual‑visibility set contains vertices from both sides of a strong bridge, the bridge’s tail must be the only vertex from the source side in the set, and the bridge’s head must be the only vertex from the sink side. This “bottleneck” effect can dramatically limit the size of a mutual‑visibility set.

However, Proposition 2.6 demonstrates that the number of strong bridges β(D) and μ(D) are not linearly related. The authors give two contrasting families: (i) directed cycles Cₙ where every edge is a strong bridge (β(Cₙ)=n) but μ(Cₙ)=2, and (ii) a construction consisting of two large complete digraphs linked by two opposite bridges, where β(D)=2 yet μ(D) can be made arbitrarily large by choosing large cliques. Thus, strong bridges provide useful local constraints but do not globally bound μ(D).

The paper proceeds to compute exact μ(D) values for several fundamental digraph classes. For any directed acyclic graph (DAG) the mutual‑visibility number is μ(D)=1, because the absence of directed cycles precludes any pair of vertices from being mutually reachable. For directed cycles Cₙ (n ≥ 3) the authors prove μ(Cₙ)=2: although each pair of vertices is mutually reachable, the cyclic structure forces any shortest path between two vertices to contain at least one intermediate vertex from the set, limiting the set size to two.

The most striking positive result concerns tournaments. Using properties of Paley tournaments—a class of regular, highly symmetric tournaments—the authors show that μ(T) grows linearly with the order of the tournament. In these digraphs every pair of vertices is mutually reachable, and the dense arc structure allows the construction of large mutual‑visibility sets that avoid internal interference. This demonstrates that, unlike DAGs and directed cycles, certain dense digraphs can support arbitrarily large mutual‑visibility sets.

From an algorithmic perspective, the authors present a verification algorithm that checks whether a given vertex set S is a mutual‑visibility set in O(|S|·(|V|+|A|)) time by computing all‑pairs shortest paths restricted to the complement of S. They then prove that the optimization problem—determining μ(D) for an arbitrary digraph—is NP‑hard via a reduction from a known hard problem (details omitted in the excerpt). This hardness result justifies the focus on special graph families and suggests the need for approximation or parameterized algorithms.

In summary, the paper makes several contributions: (1) a rigorous definition of mutual visibility for directed graphs; (2) structural theorems linking μ(D) to SCCs and strong bridges; (3) exact values for DAGs (μ=1) and directed cycles (μ=2); (4) a linear lower bound for tournaments, especially Paley tournaments; (5) a polynomial‑time verification procedure; and (6) an NP‑hardness proof for the general optimization problem. These results broaden the theoretical foundation of visibility concepts in directed networks and have potential applications in routing, robot swarm coordination, and communication systems where link directionality matters.