A Degree Threshold for Independent Domination in Generalized Prisms
Abstract
We study per-colour independent (k)-rainbow domination and its connection with independent domination in generalized prisms. Building on the known prism identity and the trivial regime above the maximum degree, we focus on the boundary case where the number of colours equals the maximum degree. For every fixed (k\ge 3), we prove that the decision problem remains NP-complete even on a highly restricted class of graphs: (C_4)-free, bipartite, ((k,2))-biregular subdivision graphs arising from simple (k)-regular graphs. The reduction gives an exact correspondence between optimal rainbow-independent dominating functions on the subdivision graph and proper (k)-edge-colourings of the original graph. We also introduce an excess parameter measuring how far the domination number lies above its natural lower bound. For cubic graphs, this excess coincides with the classical edge-colouring degree and therefore with standard resistance parameters for subcubic graphs. These results reveal a sharp one-unit threshold: above the maximum degree the problem becomes trivial for every graph, while at the boundary NP-hard instances already occur within a very narrow structural family.