On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results

Juan A. Rodríguez-Velázquez, Henning Fernau

Published 2013-09-09, updated 2013-10-12Version 2

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency metric dimension and the local metric dimension have been introduced. We combine these variants and introduce the local adjacency metric dimension. We show that the (local) metric dimension of the corona product of a graph of order $n$ and some non-trivial graph $H$ equals $n$ times the (local) adjacency metric dimension of $H$. This strong relation also enables us to infer computational hardness results for computing the (local) metric dimension, based on according hardness results for (local) adjacency metric dimension that we also provide. We also study combinatorial properties of the strong product of graphs and emphasize the role different types of twins play in determining in particular the adjacency metric dimension of a graph.