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

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.

No reviews yet.

No comments yet.