The idea of enumeration algorithms with polynomial delay is to polynomially bound the running time between any two subsequent solutions output by the enumeration algorithm. While it is open for more than four decades if all minimal dominating sets of a graph can be enumerated in output-polynomial time, it has recently been proven that pointwise-minimal Roman dominating functions can be enumerated even with polynomial delay. The idea of the enumeration algorithm was to use polynomial-time solvable extension problems. We use this as a motivation to prove that also two variants of Roman dominating functions studied in the literature, named perfect and unique response, can be enumerated with polynomial delay. This is interesting since Extension Perfect Roman Domination is W[1]-complete if parameterized by the weight of the given function and even W[2]-complete if parameterized by the number vertices assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability of extension problems and enumerability with polynomial delay tend to go hand-in-hand. We achieve our enumeration result by constructing a bijection to Roman dominating functions, where the corresponding extension problem is polynomimaltime solvable. Furthermore, we show that Unique Response Roman Domination is solvable in polynomial time on split graphs, while Perfect Roman Domination is NP-complete on this graph class, which proves that both variations, albeit coming with a very similar definition, do differ in some complexity aspects. This way, we also solve an open problem from the literature.

翻译：关于多项式延迟枚举算法的核心思想是，在枚举算法输出的任意两个连续解之间，运行时间需被多项式界限约束。尽管四十多年来，图的所有极小支配集是否能在输出多项式时间内被枚举仍是开放问题，但最近已证明，逐点极小罗马支配函数甚至可以通过多项式延迟进行枚举。该枚举算法的思想是利用多项式时间可解的扩展问题。我们以此为契机证明，文献中研究的罗马支配函数的两种变体——完美支配与唯一响应——也能通过多项式延迟进行枚举。这一结果具有理论意义，因为正如我们所证明的，当参数化为给定函数的权重时，完美罗马支配扩展问题是W[1]-完备的；若参数化预解中赋值为0的顶点数量，则是W[2]-完备的。否则，扩展问题的可解性与多项式延迟枚举性通常相互关联。我们通过构造一个到罗马支配函数的双射来实现枚举结果，该双射对应的扩展问题是多项式时间可解的。此外，我们证明唯一响应罗马支配在分裂图上可在多项式时间内求解，而完美罗马支配在此图类上是NP-完备的，这表明尽管这两种变体定义相似，但在某些复杂度方面存在差异。通过这种方式，我们还解决了文献中的一个开放问题。