Problems from metric graph theory such as Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact on the field of parameterized complexity by being the first problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. More specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to hypergraph dualization, arguably one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in different works this last decade: for which vertex (or edge) set graph property $\Pi$ is the enumeration of minimal (or maximal) subsets satisfying $\Pi$ equivalent to hypergraph dualization? As only very few properties are known to fit within this context -- namely, properties related to minimal domination -- our results make significant progress in characterizing such properties, and provide new angles of approach for tackling hypergraph dualization. In a second step, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show these cases to be mainly tractable.

翻译：度量图论中的问题，如度量维数、测地集和强度量维数，最近对参数化复杂性领域产生了重要影响，成为NP中首批在树宽上允许双指数下界的问题，后者甚至在顶点覆盖数上也是如此。我们首次研究了这些问题的最小解集枚举，并表明它们在枚举中也具有重要价值。具体而言，我们证明了在图中枚举最小分辨集和在分裂图中枚举最小测地集等价于超图对偶化——这可以说是算法枚举领域最重要的开放问题之一。这为过去十年不同研究中出现的一个问题提供了两个新的自然实例：对于哪些顶点（或边）集图性质$\Pi$，满足$\Pi$的最小（或最大）子集枚举等价于超图对偶化？由于已知仅极少数性质（即与最小支配相关的性质）符合这一背景，我们的结果在刻画此类性质方面取得了显著进展，并为解决超图对偶化提供了新的视角。其次，我们考虑了我们的归约不适用的情况，即无长诱导路径的图，并表明这些情况主要是易处理的。