Edge metric basis and its fault tolerance over certain interconnection networks

The surveillance of elements in an interconnection network is a classical problem in computer engineering. In addition, it is a problem closely related to uniquely identifying the elements of the network, which is indeed a classical distance-related problem in graph theory. This surveillance can be considered for different styles of elements in the network. The classical version centers the attention on the nodes, while some recent variations of it consider monitoring also the edges or both, vertices and edges at the same time. The first style gave rise to graph structures, called edge resolving set and edge metric basis, which is used to uniquely identify the edges of a given network by means of distance vectors. A vertex x in a graph G uniquely recognizes (resolves or identifies) two edges e and f in G if $d_G[e,x]\ne d_G[f,x]$, where $d_G[e,x]$ stands for the distance between a vertex x and an edge e of G. A set S with the smallest number of vertices, such that every couple of edges is uniquely recognized by a minimum of one vertex in S, is an edge metric basis, and the edge metric dimension refers to the cardinality of such S. Fault tolerance of a working system is the ability of such a system to keep functioning even if one of its parts stops working properly. The fault tolerance property of the edge metric basis is considered in this work. This results in a concept called fault-tolerant edge metric basis. That is, an edge metric basis S of a graph G is fault-tolerant if every pair of edges of G are resolved by a minimum of two vertices in S, and the minimum possible cardinality of such sets is coined as the fault-tolerant edge metric dimension of G. In this work, we present bounds for the edge metric dimension of graphs and its fault tolerance version. In addition, we investigate these parameters for butterfly, Beneš and fractal cubic networks, and found the exact value for their (fault-tolerant) edge metric dimensions.