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For an ordered subset $W = \{w_1, w_2,\dots w_k\}$ of vertices and a vertex$u$ in a connected graph $G$, the representation of $u$ with respect to $W$ isthe ordered $k$-tuple $ r(u|W)=(d(v,w_1), d(v,w_2),\dots,$ $d(v,w_k))$, where$d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$is a local metric generator for $G$ if every two adjacent vertices of $G$ havedistinct representations. A minimum local metric generator is called a\emph{local metric basis} for $G$ and its cardinality the \emph{local metricdimension} of G. We show that the computation of the local metric dimension ofa graph with cut vertices is reduced to the computation of the local metricdimension of the so-called primary subgraphs. The main results are applied tospecific constructions including bouquets of graphs, rooted product graphs,corona product graphs, block graphs and chain of graphs.

Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs