Complexity of Minimum Selective Subset on unit disk graphs for a constant number of colors

Determine whether the Minimum Selective Subset problem on unit disk graphs is NP-hard when the number of colors c is a fixed constant (e.g., c=2), given that NP-completeness has been established when c is unbounded.

Background

The paper proves that the Minimum Selective Subset (MSS) problem is NP-complete on unit disk graphs when the number of colors c is arbitrary, via a reduction from Dominating Set. It also provides a polynomial-time approximation scheme (PTAS) for unit disk graphs without requiring a geometric representation.

However, the reduction establishing NP-completeness relies on an unbounded number of colors. Whether the hardness persists when c is a fixed constant remains unresolved, motivating a precise complexity classification for this restricted setting.