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.

翻译：来自度量图论的问题，如度量维数、测地集和强度量维数，最近在参数化复杂性中产生了重大影响，成为NP中首批已知的问题，在树宽上具有双指数下界，甚至在后者的顶点覆盖数上，假设指数时间假说。我们启动了枚举这些问题的最小解集的研究，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枚举超图中的最小横贯（记为Trans-Enum），其全多项式时间可解性是算法枚举中最重要的开放问题之一。这为近年来出现的一个问题提供了两个新的自然示例：对于哪些顶点（或边）集图性质Π，枚举满足Π的最小（或最大）子集等价于Trans-Enum？由于已知只有极少数性质属于这一范畴——即与最小支配相关的性质——我们的结果在刻画这些性质方面取得了显著进展，并为处理Trans-Enum提供了新的视角。相比之下，我们观察到最小强分辨集可以以多项式延迟枚举。此外，我们考虑了我们的归约不适用的情况，即没有长诱导路径的图，并展示了与部分解的枚举和扩展相关的正面和负面结果。