NP-hardness for subset Gray coding problems without independent involutions

Establish NP-hardness for Gray coding existence problems on subset families whose flip operations do not support independent involutions (i.e., operations that do not allow multiple local changes to be made and then unmade independently), for which NP-hardness could not be shown using grid-graph or hypercube reductions.

Background

The paper develops NP-completeness results for Gray coding existence problems using reductions that embed hypercubes or grid graphs into flip graphs of various combinatorial objects. These reductions rely on having many independent involutive flips, enabling local changes to be made and unmade independently, which is characteristic of hypercube-based constructions.

In the concluding remarks, the authors explicitly note that they were unable to establish NP-hardness for certain subset-based Gray coding problems where the available flip operations do not support independent involutions. This identifies a gap in the current techniques and highlights a class of problems whose complexity remains unresolved within the paper’s framework.