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Fault-tolerance in resolvability. (English) Zbl 1197.05041
Summary: A subset $$W= \{w1,\dots,w_k\}A$$ of vertices of a graph $$G$$ is called a resolving set for $$G$$ if for every two distinct vertices $$x,y\in V(G)$$ there is a vertex $$w_i\in W$$ such that $$d(x,w_i)\neq d(y,w_i)$$. A resolving set $$W$$ for $$G$$ is fault-tolerant if $$W\setminus\{w\}$$, 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 $$\beta'(G)$$. A $$k$$-partition $$\Pi=\{S_1,S_2,\dots, S_k\}$$ of $$V(G)$$ is said to be fault-tolerant resolving if the codes $$c_\Pi(v)$$, $$v\in V(G)$$, differ by at least two coordinates where $$c_\Pi(v)= (d(v,S_1),d(v,S_2),\dots,d(v,S_k))$$. 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 $${\mathcal P}(G)$$. In this paper, we study fault-tolerant metric dimension and fault-tolerant partition dimension of graphs and their relationship. We show that every pair $$a, b$$ $$(a\neq b-1)$$ of positive integers with $$5\leq a\leq b$$ 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 $$G$$.

##### MSC:
 05C12 Distance in graphs