  Problems from metric graph theory like Metric Dimension, Geodetic Set, and
Strong Metric Dimension have recently had a strong impact in parameterized
complexity by being the first known problems in NP to admit double-exponential
lower bounds in the treewidth, and even in the vertex cover number for the
latter, assuming the Exponential Time Hypothesis. We initiate the study of
enumerating minimal solution sets for these problems and show that they are
also of great interest in enumeration. Specifically, we show that enumerating
minimal resolving sets in graphs and minimal geodetic sets in split graphs are
equivalent to enumerating minimal transversals in hypergraphs (denoted
Trans-Enum), whose solvability in total-polynomial time is one of the most
important open problems in algorithmic enumeration. This provides two new
natural examples to a question that emerged in recent works: for which vertex
(or edge) set graph property $\Pi$ is the enumeration of minimal (or maximal)
subsets satisfying $\Pi$ equivalent to Trans-Enum? As very few properties are
known to fit within this context -- namely, those related to minimal domination
-- our results make significant progress in characterizing such properties, and
provide new angles to approach Trans-Enum. In contrast, we observe that minimal
strong resolving sets can be enumerated with polynomial delay. Additionally, we
consider cases where our reductions do not apply, namely graphs with no long
induced paths, and show both positive and negative results related to the
enumeration and extension of partial solutions.
