Characterize input-graph families admitting fixed-parameter tractable dilation augmentation

Characterize the families of input graphs \mathcal{G} 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 input graph G determines the existing weighted edges, while the metric is supplied independently through the distances between vertex pairs. Structural restrictions on G can substantially affect the complexity of Dilation t-Augmentation: the paper proves tractability for several sparse or bounded-degree classes and hardness for other settings.

A general characterization of graph families yielding fixed-parameter tractability remains unresolved. The explicitly posed question seeks to identify which structural properties of G suffice for efficient parameterized algorithms.