Finite/cofinite regimes not covered by the rank-width lower bound

Determine whether the quadratic 2^{O(k^2)} rank-width dependence is necessary for the three remaining finite/cofinite \((\sigma,\rho)\)-set minimization regimes with \(0\notin\rho\): finite–finite, cofinite–finite, and the finite–cofinite pairs not satisfying \(0\in\sigma\) and \(1\in\rho\).

Background

The paper proves matching quadratic rank-width lower bounds when $\rho$ is cofinite and either $\sigma$ is cofinite or $0\in\sigma$ and $1\in\rho$. These results settle the cofinite–cofinite minimization regime and a subfamily of the finite–cofinite regime.

Three finite/cofinite regimes remain incompletely covered. The finite–finite case lacks both large-clique validation and cofinite padding; the cofinite–finite case requires exact control of the unselected vertices; and the remaining finite–cofinite pairs require control of selected vertices without either a large clique or the independent-solution assumptions used in the paper.