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

Abstract

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 $W$ is the 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 $G$ have distinct representations. A minimum local metric generator is called a \emph{local metric basis} for $G$ and its cardinality the \emph{local metric dimension} of G. We show that the computation of the local metric dimension of a graph with cut vertices is reduced to the computation of the local metric dimension of the so-called primary subgraphs. The main results are applied to specific constructions including bouquets of graphs, rooted product graphs, corona product graphs, block graphs and chain of graphs.