Computing the metric dimension of a graph from primary subgraphs
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by
D. Kuziak, J. A. Rodríguez-Velázquez, I. G. Yero
2015
Abstract
Let G be a connected graph. Given an ordered set W = {w_1, w_2,...
w_k}⊆ V(G) and a vertex u∈ V(G), the representation of u with
respect to W is the ordered k-tuple (d(u,w_1), d(u,w_2),...,
d(u,w_k)), where d(u,w_i) denotes the distance between u and w_i. The
set W is a metric generator for G if every two different vertices of G
have distinct representations. A minimum cardinality metric generator is called
a metric basis of G and its cardinality is called the metric
dimension of G. It is well known that the problem of finding the metric
dimension of a graph is NP-Hard. In this paper we obtain closed formulae for
the metric dimension of graphs with cut vertices. The main results are applied
to specific constructions including rooted product graphs, corona product
graphs, block graphs and chains of graphs.
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