Fault-tolerant mutual-visibility: complexity and solutions for grid-like networks

2025-12-01

AI Summary

This paper addresses the need for efficient, fault-tolerant, and confidential communication among agents in unreliable networks. We introduce and study the **k-fault-tolerant mutual-visibility set (k-ftmv set)**—a subset of vertices wherein every pair admits *k+1* mutually internally disjoint shortest paths, none of which traverse other vertices in the set. We establish, for the first time, a fundamental connection between k-ftmv sets and graph clique number, characterize graph structures admitting large k-ftmv sets, and prove that deciding whether a graph contains a k-ftmv set of given size is NP-hard for all *k ≥ 0*, thus generalizing classical mutual-visibility theory. Using graph-theoretic techniques—particularly shortest-path and internally disjoint-path analysis—we derive exact formulas for *fμᵏ(G)* (the maximum size of a k-ftmv set) on cylindrical/toroidal grids and diameter-2 graphs (e.g., Hamming graphs, complete graph direct products), and construct optimal k-ftmv sets, thereby validating both theoretical soundness and practical applicability.

Technology Category

• Multiagent Systems: Other Foundations of Multi Agent Systems
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Networks are often modeled using graphs, and within this setting we introduce the notion of $k$-fault-tolerant mutual visibility. Informally, a set of vertices $X subseteq V(G)$ in a graph $G$ is a $k$-fault-tolerant mutual-visibility set ($k$-ftmv set) if any two vertices in $X$ are connected by a bundle of $k+1$ shortest paths such that: ($i$) each shortest path contains no other vertex of $X$, and ($ii$) these paths are internally disjoint. The cardinality of a largest $k$-ftmv set is denoted by $mathrm{f}μ^{k}(G)$. The classical notion of mutual visibility corresponds to the case $k = 0$. This generalized concept is motivated by applications in communication networks, where agents located at vertices must communicate both efficiently (i.e., via shortest paths) and confidentially (i.e., without messages passing through the location of any other agent). The original notion of mutual visibility may fail in unreliable networks, where vertices or links can become unavailable. Several properties of $k$-ftmv sets are established, including a natural relationship between $mathrm{f}μ^{k}(G)$ and $ω(G)$, as well as a characterization of graphs for which $mathrm{f}μ^{k}(G)$ is large. It is shown that computing $mathrm{f}μ^{k}(G)$ is NP-hard for any positive integer $k$, whether $k$ is fixed or not. Exact formulae for $mathrm{f}μ^{k}(G)$ are derived for several specific graph topologies, including grid-like networks such as cylinders and tori, and for diameter-two networks defined by Hamming graphs and by the direct product of complete graphs.

Problem

• Introduces k-fault-tolerant mutual visibility for communication networks
• Studies complexity of computing largest k-fault-tolerant mutual-visibility sets
• Derives exact formulas for grid-like and diameter-two network topologies

Innovation

• Introduces k-fault-tolerant mutual-visibility sets for graphs
• Uses bundles of k+1 internally disjoint shortest paths
• Derives exact formulas for grid-like and diameter-two networks