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Resolvability in graphs and the metric dimension of a graph. (English) Zbl 0958.05042
For an ordered set $$W=\{w_{1},\ldots ,w_{k}\}$$ of vertices in a connected graph $$G$$ and a vertex $$v$$ of $$G$$, the metric representation of $$v$$ with respect to $$W$$ is the $$k$$-vector $$r(v|W)=(d(v,w_{1}),\ldots ,d(v,w_{k}))$$. The set $$W$$ is said to be a resolving set for $$G$$ if $$r(u|W)=r(v|W)$$ implies that $$u=v$$ for all pairs $$u$$, $$v$$ of vertices of $$G$$. The metric dimension $$\dim(G)$$ of $$G$$ is the minimum cardinality of a resolving set for $$G$$. In this paper bounds on $$\dim(G)$$ are presented in terms of the order and the diameter of $$G$$. All connected graphs of order $$n$$ having dimension 1, $$n-2$$, or $$n-1$$ are determined and a new proof for the dimension of a tree is also presented. From this result sharp bounds on the metric dimension of unicyclic graphs are established. It is shown that $$\dim(H)\leq \dim(H\times K_{2})\leq \dim(H)+1$$ for every connected graph $$H$$. Moreover, it is shown that for every positive real number $$\varepsilon$$, there exists a connected graph $$G$$ and a connected induced subgraph $$H$$ of $$G$$ such that $$\dim(G)/\dim(H)<\varepsilon$$.

##### MSC:
 05C12 Distance in graphs 05C05 Trees
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##### References:
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