Unique fault-tolerance resolvability codes for certain generalized convex polytopes with applications

Abstract

For a non-trivial connected graph [Formula: see text], an ordered subset [Formula: see text] of vertices [Formula: see text] any pair of different vertices [Formula: see text], if [Formula: see text] for some [Formula: see text]. Such a set [Formula: see text] is said to be a resolving set (RS) for [Formula: see text] and the smallest cardinality of [Formula: see text] is called the [Formula: see text] [Formula: see text] of [Formula: see text]. A RS [Formula: see text] for [Formula: see text] is said to be fault-tolerant resolving set (FTRS) if the property of resolving [Formula: see text] holds by [Formula: see text] for every [Formula: see text] in [Formula: see text]. The minimum cardinality of such a set [Formula: see text] is called the fault-tolerant metric dimension (FTMD) of [Formula: see text]. It is generally denoted by [Formula: see text]. In this paper, the notion of FTRS and that of FTMD have been determined for two complex families of infinite convex polytope graphs, which are obtained by joining certain copies of the prism graph together with an antiprism graph. For these two families of planar graph (viz., [Formula: see text] & [Formula: see text]), we investigate several properties as well as we compare their metric and FTMD.