The Maximum Mutual Visibility Set on a Cactus Graph and the Self-stabilizing Constructions

Abstract

Given a graph $G=(V,E)$, let $S$ ($\subseteq V$) be a set of vertices. Two vertices are \emph{mutually visible} if there exists a shortest path in $G$ between them that does not contain any other vertex of $S$. A set $S$ is a \emph{Mutual Visibility Set} (\MVS) if every pair of vertices in $S$ is mutually visible. The concept of \MVS s in graphs has attracted significant attention since its introduction, as it provides an important structural property of graphs. However, determining a maximum \MVS\ in general graphs is computationally intractable; the decision problem of whether a graph admits an \MVS\ of size at least $k$ has been shown to be \emph{NP-complete}. Thus, prior work has focused on finding maximal \MVS s or restricting attention to specific graph classes. Cactus graphs form a fundamental low-treewidth class, yet the maximum \MVS\ problem for this class remains open. In this paper, we first determine the size of maximum \MVS~in cactus graphs, and introduce two self-stabilizing algorithms that construct such sets. The first algorithm uses a single BFS tree and stabilizes in $O(D)$ rounds with $O(\log n)$ bits per process on average; the second one uses parallel BFS trees and stabilizes in $O(|C_{\max}|+|T_{\max}|)$ rounds, which we show to be asymptotically tight as a function of these two parameters, even on graphs where $|C_{\max}|+|T_{\max}| = o(D)$.