Fault-tolerance in resolvability
Abstract
A subset W = {w1,⋯ ,wk} of vertices of a graph G is called a resolving set for G if for every two distinct vertices x,y ∈ V(G) there is a vertex wi ∈ W such that d(x, wi) ≠ d(y, wi). 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 β′{G). A kpartition Π = {S1, S2,⋯, Sk} of V(G) is said to be fault-tolerant resolving if the codes cΠ(u), u ∈ V(G), differ by at least two coordinates where cΠ(u) = (d(u,S1),d(u, S2),⋯ ,d(u, Sk))-The minimum k for which there is a fault-tolerant resolving k-partition of V(G) is called the fault-tolerant partition dimension of G, denoted by P(G). In this paper, we study fault-tolerant metric dimension and fault-tolerant partition dimension of graphs and their relationship. We show that every pair a, 6(a ≠ 6-1) of positive integers with 5 ≥ a ≥ b is realizable as the fault-tolerant metric dimension and the fault-tolerant partition dimension for some connected graphs. Also, bounds on maximum order of a graph G are presented in terms of diameter, fault-tolerant metric dimension and fault-tolerant partition dimension of G.
