Rank-width lower bound for Perfect Dominating Set

Establish a matching rank-width lower bound for Perfect Dominating Set, whose feasible solutions require every unselected vertex to have exactly one selected neighbor, while preserving an interface of rank O(k) for instances encoding k^2 assignment bits.

Background

The paper’s lower-bound constructions handle domination constraints that distinguish zero selected neighbors from one or more selected neighbors by using cofinite padding. Perfect Dominating Set instead requires an exact-one condition for every unselected vertex, so the existing padding strategy does not apply directly.

A successful construction would need an exact-one checker that prevents multiple selected assignment neighbors without exposing quadratic assignment information across a cut and thereby exceeding the required O(k) rank bound. The paper notes that the general 2^{O(k^2)} upper bound is known, but no matching lower bound is currently known.