AI 中文总结
研究固定整数k下给定图G的精确支配数问题，通过从3SAT进行多项式时间归约，结合更简单归约，证明了对于k≥3，Exact-k-DNP是DP完全的，弥合了k = 3和4时的分类差距，完成固定值分类。
AI 中文摘要
精确支配数问题是对于固定整数k，判断给定图G是否满足dom(G)=k。Riege和Rothe证明了对于每个固定k≥5，该问题是DP完全的，而k = 3和k = 4的情况仍未解决。本文弥合了这一分类差距。主要方法是从3SAT进行多项式时间归约，在可满足情况下输出图的支配数为4，不可满足情况下为2，且不会产生支配数为3的图。结合一个更简单的三对二归约，得到了Exact-3-DNP和Exact-4-DNP的DP完全性。证明是构造性的，给出了明确的图小工具，其局部支配约束编码真值赋值和子句满足情况。反之，健全性论证表明任何足够大的支配划分都强制了预期的一致性条件，从而产生一个满足赋值。因此，对于每个固定k≥3，Exact-k-DNP是DP完全的，完成了从k = 3开始的固定值分类。
英文摘要
The exact domatic-number problem asks, for a fixed integer k, whether a given graph G satisfies dom(G) = k. Riege and Rothe proved DP-completeness for every fixed k >= 5, while the cases k = 3 and k = 4 remained open. We close this classification gap. The main ingredient is a polynomial-time reduction from 3SAT whose output graphs have domatic number 4 in the satisfiable case and domatic number 2 in the unsatisfiable case; in particular, the reduction never produces a graph of domatic number 3. This directly realizes the route suggested by Riege and Rothe for closing the remaining cases. Together with a simpler three-versus-two reduction, this yields DP-completeness of Exact-3-DNP and Exact-4-DNP. The proofs are constructive and give explicit graph gadgets whose local domination constraints encode truth assignments and clause satisfaction. The soundness arguments show conversely that any sufficiently large domatic partition enforces the intended consistency conditions and therefore yields a satisfying assignment. Consequently, Exact-k-DNP is DP-complete for every fixed k >= 3, completing the fixed-value classification from k = 3 onward.
