On the unicyclic graphs having vertices that belong to all their (strong) metric bases

TL;DR

This paper investigates unicyclic graphs to classify those with vertices that are mandatory in all (strong) metric bases, providing new characterizations and revealing differences between metric and strong metric bases.

Contribution

It offers the first classification of unicyclic graphs based on the number of basis forced vertices and characterizes conditions for their existence.

Findings

Unicyclic graphs can have at most two basis forced vertices.

Unicyclic graphs can have an arbitrary number of strong basis forced vertices.

Characterizations of unicyclic graphs with respect to these vertices are provided.

Abstract

A metric basis in a graph $G$ is a smallest possible set $S$ of vertices of $G$, with the property that any two vertices of $G$ are uniquely recognized by using a vector of distances to the vertices in $S$. A strong metric basis is a variant of metric basis that represents a smallest possible set $S'$ of vertices of $G$ such that any two vertices $x,y$ of $G$ are uniquely recognized by a vertex $v\in S'$ by using either a shortest $x-v$ path that contains $y$, or a shortest $y-v$ path that contains $x$. Given a graph $G$, there exist sometimes some vertices of $G$ such that they forcedly belong to every metric basis or to every strong metric basis of $G$. Such vertices are called (resp. strong) basis forced vertices in $G$. It is natural to consider finding them, in order to find a (strong) metric basis in a graph. However, deciding about the existence of these vertices in arbitrary graphs is in general an NP-hard problem, which makes desirable the problem of searching for (strong) basis forced vertices in special graph classes. This article centers the attention in the class of unicyclic graphs. It is known that a unicyclic graph can have at most two basis forced vertices. In this sense, several results aimed to classify the unicyclic graphs according to the number of basis forced vertices they have are given in this work. On the other hand, with respect to the strong metric bases, it is proved in this work that unicyclic graphs can have as many strong basis forced vertices as we would require. Moreover, some characterizations of the unicyclic graphs concerning the existence or not of such vertices are given in the exposition as well.