Fault-Tolerant Metric and Partition Dimension of Graphs

A set W of vertices in a graph G is called a resolving set for G if for every pair of distinct vertices u and v of G there exists a vertex w is an element of W such that the distance between u and w is different from the distance between v and w. The cardinality of a minimum resolving set is called the metric dimension of C, denoted by beta(G). A resolving set W' for G is fault-tolerant if W'\w}, for each w in W', is also a resolving set and the fault-tolerant metric dimension of G is the minimum cardinality of such a set, denoted by beta'(G). We characterize all the graphs G such that beta'(C) - beta(G) = 1. A k-partition Pi = {S-1, S-2, ... , S-k} of V(C) is resolving if for every pair of distinct vertices u, v in C, there is a set S, in II so that the minimum distance between u and a vertex of S-i is different from the minimum distance between v and a vertex of S-i. A resolving partition Pi is fault-tolerant if for every pair of distinct vertices u and v in V(G), there are at least two sets S-i, S-j in Pi so that the minimum distance between u and a vertex of S-i and a vertex of S-j is different from the minimum distance between v and a vertex of S-i and a vertex of S-j. The cardinality of a minimum fault-tolerant resolving partition is called the fault-tolerant partition dimension, denoted by (G). In this paper, we show that every pair a, b of positive integers with b >= 6 and inverted left perpendicular b/2 inverted right perpendicular + 1 <= a <= b - 2 is realizable as the fault-tolerant metric dimension and the fault-tolerant partition dimension of some connected graphs. Also, we show that P(G) = n if and only if G = K-n or G = K-n - e.