Characterize metrics admitting fixed-parameter tractable dilation augmentation

Characterize the metrics for which Dilation t-Augmentation admits an algorithm with running time f(k) 00 n^{O(1)}, parameterized by the number k of added edges.

Background

Dilation t-Augmentation asks whether at most k edges can be added to an embedded graph G so that its dilation becomes at most t. The paper studies this problem primarily when the metric is induced by the shortest-path metric of an unweighted graph, and establishes fixed-parameter tractability and hardness results for several restricted graph classes.

The paper does not provide a classification of all metrics for which the problem is fixed-parameter tractable. The question is motivated by the observation that the problem is computationally hard in full generality, even when the metric is induced by a clique. It therefore asks for the boundary between tractable and intractable metric classes.