AI 中文总结
研究图中超级联盟划分的超级联盟数$C_s(G)$，利用底层阶数和超级支配数建立界限，分析其与超级支配度关系及计算复杂性，为多种标准图架构提供精确确定，证明其可任意大。
AI 中文摘要
我们引入并研究了图中超级联盟划分的结构特性，这是一个将合作资源部署与严格支配标准联系起来的新方向。基于超级支配的基本概念，超级联盟划分被定义为顶点集划分$\Upsilon = \{A_1, A_2, \ldots, A_k\}$，使得没有单个类$A_i$构成有效的超级支配集，但每个类都可以与至少一个不同的伙伴类$A_j$配对，形成并集$A_i \cup A_j$，实现对图的完全超级支配。超级联盟数用$C_s(G)$表示，代表这种划分的最大可能基数。在本文中，我们利用底层阶数和超级支配数$\gamma_{sp}(G)$为$C_s(G)$建立了一般操作界限，展示了它与超级支配度$d_{sp}(G)$的关系，分析了其计算复杂性，证明了在一般条件下它是NP完全的，并为包括路径、循环、完全图、星图、轮图和友谊配置在内的关键标准图架构提供了精确确定。我们通过证明超级联盟数可以任意大来得出结论。
英文摘要
We introduce and investigate the structural properties of super coalition partitions in graphs, a novel direction that bridges cooperative resource deployment with rigid domination criteria. Based on the foundational concept of super domination, a super coalition partition is defined as a vertex set partitioning $Υ= \{A_1, A_2, \ldots, A_k\}$ such that no single class $A_i$ constitutes a valid super dominating set, yet every class can be paired with at least one distinct partner class $A_j$ to form a union $A_i \cup A_j$ that achieves full super domination over the graph. The super coalition number, denoted by $C_s(G)$, represents the maximum possible cardinality of such a partition. In this paper, we establish general operational bounds for $C_s(G)$ using the underlying order and the super domination number $γ_{sp}(G)$, demonstrate its relation to the super domatic number $d_{sp}(G)$, analyze its computational complexity proving its NP-complete nature under general conditions, and provide exact determinations for key standard graph architectures including paths, cycles, complete graphs, stars, wheels, and friendship configurations. We conclude by proving that the super coalition number can grow arbitrarily large.
