On Vertices Contained in All or in No Metric Basis

TL;DR

This paper investigates vertices that are always or never part of any metric basis in a graph, providing structural properties, bounds, construction methods, and complexity results for related decision problems.

Contribution

It introduces the concept of basis forced vertices, explores their properties in various graph classes, and establishes complexity results for related decision problems.

Findings

Bounds for maximum edges in graphs with basis forced vertices

Construction methods for sparse graphs with basis forced vertices

Deciding vertex inclusion in all or no metric bases is co-NP-hard or NP-hard

Abstract

A set $R \subseteq V(G)$ is a resolving set of a graph $G$ if for all distinct vertices $v,u \in V(G)$ there exists an element $r \in R$ such that $d(r,v) \neq d(r,u)$. The metric dimension $\dim(G)$ of the graph $G$ is the minimum cardinality of a resolving set of $G$. A resolving set with cardinality $\dim(G)$ is called a metric basis of $G$. We consider vertices that are in all metric bases, and we call them basis forced vertices. We give several structural properties of sparse and dense graphs where basis forced vertices are present. In particular, we give bounds for the maximum number of edges in a graph containing basis forced vertices. Our bound is optimal whenever the number of basis forced vertices is even. Moreover, we provide a method of constructing fairly sparse graphs with basis forced vertices. We also study vertices which are in no metric basis in connection to cut-vertices and pendants. Furthermore, we show that deciding whether a vertex is in all metric bases is co-NP-hard, and deciding whether a vertex is in no metric basis is NP-hard.