Fault-tolerant edge metric dimension of certain families of graphs

Abstract

Let WE = {w1, w2, . . ., wk } be an ordered set of vertices of graph G and let e be an edge of G. Suppose d(x, e) denotes distance between edge e and vertex x of G, defined as d(e, x) = d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) d(e2, x). The representation r(e | WE ) of e with respect to WE is the k-tuple (d(e, w1 ), d(e, w2 ), . . ., d(e, wk )). If distinct edges of G have distinct representation with respect to WE, then WE is called edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). In this paper, we initiate the study of fault-tolerant edge metric dimension. Let ẂE be edge metric generator of graph G, then ẂE is called fault-tolerant edge metric generator of G if ẂE \ {v} is also an edge metric generator of graph G for every v ∈ ẂE . A fault-tolerant edge metric generator of minimum cardinality is a fault-tolerant edge metric basis for graph G, and its cardinality is called fault-tolerant edge metric dimension of G. We also computed the fault-tolerant edge metric dimension of path, cycle, complete graph, cycle with chord graph, tadpole graph and kayak paddle graph.