Fixed-parameter tractability of Minimum Selective Subset parameterized by the number of colors on unit disk graphs

Ascertain whether the Minimum Selective Subset problem on unit disk graphs is fixed-parameter tractable when parameterized by the number of colors c.

Background

Following the open complexity question for constant c on unit disk graphs, a natural parameterized question arises: if the problem's difficulty depends strongly on the number of colors, it is meaningful to ask whether an algorithm whose running time is f(c)·n^O(1) exists.

The paper establishes hardness and approximation results for various graph classes and provides a PTAS for unit disk graphs, but does not settle the parameterized complexity with respect to c.