AI 中文总结
研究图的支配临界性的复杂性，确定规定支配数和不受限制时识别支配临界图的情况，DomCrit_1和DomCrit_2可多项式时间判定，k >= 3时DomCrit_k是DP完全的，无限制问题证明了DP硬度和属于Theta_2^p。
AI 中文摘要
图G的支配数dom(G)是其顶点集划分中支配集的最大数量。若删除任何一条边都会降低其支配数，则该图是支配临界的。我们确定了在规定支配数和支配数不受限制这两种情况下，识别支配临界图的复杂性。问题DomCrit_1和DomCrit_2是多项式时间可判定的；特别地，DomCrit_2恰好由非平凡星的非空不相交并集组成。相比之下，对于每个固定整数k >= 3，问题DomCrit_k在多项式时间多一归约下是DP完全的。目标值为三时的硬度证明使用了一种开关构造，该构造从边最小的3不可着色性进行归约，并控制删除构造图每条边的效果。然后通过团添加将目标值为三的结果提升到每个更大的固定目标值。对于无限制的识别问题，我们证明了其DP硬度和属于Theta_2^p。
英文摘要
The domatic number dom(G) of a graph G is the maximum number of dominating sets in a partition of its vertex set. A graph is domatically critical if deleting any edge lowers its domatic number. We determine the complexity of recognizing domatically critical graphs both when the domatic number is prescribed and when it is unrestricted. The problems DomCrit_1 and DomCrit_2 are polynomial-time decidable; in particular, DomCrit_2 consists precisely of the nonempty disjoint unions of nontrivial stars. In contrast, for every fixed integer k >= 3, the problem DomCrit_k is DP-complete under polynomial-time many-one reductions. The hardness proof at target value three uses a switch construction that reduces from edge-minimal 3-uncolorability and controls the effect of deleting every edge of the constructed graph. Clique addition then lifts the target-three result to every larger fixed target value. For the unrestricted recognition problem, we prove DP-hardness and membership in Theta_2^p.
Comments28 pages
