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

Serafino Cicerone; Gabriele Di Stefano; Sandi Klavžar; Gang Zhang

Paper

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

Abstract

Networks are often modeled using graphs, and within this setting we introduce the notion of

-fault-tolerant mutual visibility. Informally, a set of vertices

in a graph

is a

-fault-tolerant mutual-visibility set (

-ftmv set) if any two vertices in

are connected by a bundle of

shortest paths such that: (

) each shortest path contains no other vertex of

, and (

) these paths are internally disjoint. The cardinality of a largest

-ftmv set is denoted by

. The classical notion of mutual visibility corresponds to the case

. 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

-ftmv sets are established, including a natural relationship between

and

, as well as a characterization of graphs for which

is large. It is shown that computing

is NP-hard for any positive integer

, whether

is fixed or not. Exact formulae for

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.