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

For a non-trivial connected graph Γ = Γ (V , E) , an ordered subset U of vertices r e s o l v e s any pair of different vertices k 1 , k 2 ∈ V , if d (k , k 1) ≠ d (k , k 2) for some k ∈ U. Such a set U is said to be a resolving set (RS) for Γ and the smallest cardinality of U is called the m e t r i c d i m e n s i o n of Γ. A RS U ∗ for Γ is said to be fault-tolerant resolving set (FTRS) if the property of resolving Γ holds by U ∗ − { k } for every k in U ∗ . The minimum cardinality of such a set U ∗ is called the fault-tolerant metric dimension (FTMD) of Γ. It is generally denoted by dim f t (Γ). 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., D m k & E m k ), we investigate several properties as well as we compare their metric and FTMD. [ABSTRACT FROM AUTHOR]