On Finding the Best and Worst Orientations for the Metric Dimension

Abstract

The (directed) metric dimension of a digraph D, denoted by ${{\,\mathrm{\overrightarrow{\textrm{MD}}}\,}}(D)$, is the size of a smallest subset S of vertices such that every two vertices of D are distinguished via their distances from the vertices in S. In this paper, we investigate the graph parameters ${{\,\textrm{BOMD}\,}}(G)$ and ${{\,\textrm{WOMD}\,}}(G)$ which are respectively the smallest and largest metric dimension over all orientations of G. We show that those parameters are related to several classical notions of graph theory and investigate the complexity of determining those parameters. We show that ${{\,\textrm{BOMD}\,}}(G)=1$ if and only if G is hypotraceable (that is has a path spanning all vertices but one), and deduce that deciding whether ${{\,\textrm{BOMD}\,}}(G)\le k$ is NP-complete for every positive integer k. We also show that ${{\,\textrm{WOMD}\,}}(G)\ge \alpha (G)-1$, where $\alpha (G)$ is the stability number of G. We then deduce that for every fixed positive integer k, we can decide in polynomial time whether ${{\,\textrm{WOMD}\,}}(G)\le k$. The most significant results deal with oriented forests. We provide a linear-time algorithm to compute the metric dimension of an oriented forest and a linear-time algorithm that, given a forest F, computes an orientation $D^-$ with smallest metric dimension (i.e. such that ${{\,\mathrm{\overrightarrow{\textrm{MD}}}\,}}(D^-)={{\,\textrm{BOMD}\,}}(F)$) and an orientation $D^+$ with largest metric dimension (i.e. such that ${{\,\mathrm{\overrightarrow{\textrm{MD}}}\,}}(D^+)={{\,\textrm{WOMD}\,}}(F)$).