Fault-Tolerant Metric and Partition Dimension of Graphs Muhammad Anwar Chaudhry, Imran Javaid ? , Muhammad Salman Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Pakistan Abstract. 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 ∈ 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 G, denoted by β(G). A resolving set W 0 for G is fault-tolerant if W 0 \{w}, for each w in W 0 , is also a resolving set and the fault-tolerant metric dimension of G is the minimum cardinality of such a set, denoted by β 0 (G). We characterize all the graphs G such that β 0 (G) - β(G) = 1. A k-partition Π = {S 1 ,S 2 ,...,S k } of V (G) is resolving if for every pair of distinct vertices u, v in G, there is a set S i in Π so that the minimum distance between u and a vertex of Si is different from the minimum distance between v and a vertex of S i . A resolving partition Π 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 Π 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 P (G). In this paper, we show that every pair a, b of positive integers with b ≥ 6 and d b 2 e +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 = Kn or G = Kn - e. Keywords: metric dimension, fault-tolerant metric dimension, parti- tion dimension, fault-tolerant partition dimension. 2000 Mathematics Subject Classification: 05C12 ? Corresponding author: imranjavaid45@gmail.com