AI 中文总结
该研究针对广义棱柱图，证明颜色数等于最大度时的独立(k)-彩虹控制判定问题为NP-完全，且最大度之上该问题平凡，还引入超额参数关联了三次图的边着色度与抗性参数。
AI 中文摘要
我们研究按颜色划分的独立(k)-彩虹控制问题及其与广义棱柱图中独立控制数的联系。基于已知的棱柱恒等式和最大度之上的平凡情形，我们聚焦颜色数等于最大度的边界情形。对每个固定的k≥3，我们证明该判定问题即使在高度受限的图类上仍为NP-完全问题：该图类是无(C₄)二分图，即由简单k-正则图导出的((k,2))-双正则细分图。该归约在细分图的最优彩虹独立控制函数与原图的正常k-边着色之间建立了精确对应关系。我们还引入了一个超额参数，用于衡量控制数超出其自然下界的程度。对于三次图，该超额值与经典边着色度一致，因此也与次三次图的标准抗性参数一致。这些结果揭示了一个尖锐的单位阈值：在最大度之上，该问题对所有图都变得平凡；而在边界处，已存在于极窄结构族中的NP-难实例。
英文摘要
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.
Comments12 pages
