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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 byusing a vector of distances to the vertices in $S$. A strong metric basis is avariant of metric basis that represents a smallest possible set $S'$ ofvertices of $G$ such that any two vertices $x,y$ of $G$ are uniquely recognizedby 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 existsometimes some vertices of $G$ such that they forcedly belong to every metricbasis 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 aboutthe existence of these vertices in arbitrary graphs is in general an NP-hardproblem, which makes desirable the problem of searching for (strong) basisforced vertices in special graph classes. This article centers the attention inthe class of unicyclic graphs. It is known that a unicyclic graph can have atmost two basis forced vertices. In this sense, several results aimed toclassify the unicyclic graphs according to the number of basis forced verticesthey have are given in this work. On the other hand, with respect to the strongmetric bases, it is proved in this work that unicyclic graphs can have as manystrong basis forced vertices as we would require. Moreover, somecharacterizations of the unicyclic graphs concerning the existence or not ofsuch vertices are given in the exposition as well.

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