Fault-tolerant partition resolvability of chemical chains

Abstract

An e partition X of the vertex set V (H) of a connected graph H is the collection of e number of ordered disjoint subsets of V (H), denoted as X = fX1;X2; : : : ;Xeg. The representation of a vertex u is a distance vector r(ujX) = (d(u;X1); d(u;X2); : : : ; d(u;Xe)), where d(u;Xi) is the distance of u from Xi. Any ordered e partition X is referred as resolving partition if representations of all the vertices are distinct. The smallest integer e is referred as the partition dimension of the graph. The advancement in the concept of partition dimension is fault-tolerant partition dimension where the representations are distinct at two places for each pair of vertices. In this paper, we compute the partition dimension and fault-tolerant partition dimension of some planar graphs.