Characterize tractable graph-family and metric pairs

Characterize the pairs (\mathcal{G},\mathcal{M}) consisting of an input-graph family and a metric class for which Dilation t-Augmentation admits an algorithm with running time f(k) \cdot n^{O(1)}, parameterized by the number k of added edges.

Background

The complexity of Dilation t-Augmentation depends jointly on the structure of the graph G and the metric \mathcal{M} in which it is embedded. The paper demonstrates several tractable and hard combinations, including cases where G or the metric-generating graph belongs to a restricted family.

These results do not yield a complete classification of all jointly tractable pairs of graph families and metric classes. The paper therefore explicitly asks for a characterization of such pairs.

References

For which pairs of family of input graphs and metric, $({\cal G}, \mathcal{M})$, does Dilation t-Augmentation admit an algorithm?

— Multivariate Exploration of Metric Dilation  (2501.04555 - Banik et al., 8 Jan 2025) in Section 1, Introduction