We consider structural parameterizations of the fundamental Dominating Set problem and its variants in the parameter ecology program. We give improved FPT algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). When parameterized by the deletion distance k to cluster graphs - we can find a minimum dominating set (DS) in 3^k n^{O(1)}-time. Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum dominating clique (DC) or a minimum efficient dominating set (EDS) or a minimum total dominating set (TDS). We also show that most of these variants of dominating set do not have polynomial sized kernel. Additionally, we show that when parameterized by the deletion distance k to split graphs - IDS can be solved in 2^k n^{O(1)}-time and EDS can be solved in 3^{k/2}n^{O(1)}.

翻译：我们考虑基本支配集问题及其变种在参数生态学项目中的结构参数化。针对距离簇图或分裂图至多k个顶点的图，我们改进了支配集的FPT算法，并在公认猜想下给出了下界。这类图中存在一个大小为k的顶点集（称为调制器），删除该集合后图变为簇图或分裂图。我们也将k称为删除距离（到相应图类）。当以到簇图的删除距离k为参数时，我们可以在3^k n^{O(1)}时间内找到最小支配集（DS）。在相同时间内，我们还可以找到最小独立支配集（IDS）、最小支配团（DC）、最小有效支配集（EDS）或最小总支配集（TDS）。我们还证明，这些支配集变种中的大多数不存在多项式大小的核。此外，我们证明当以到分裂图的删除距离k为参数时，IDS可在2^k n^{O(1)}时间内求解，EDS可在3^{k/2}n^{O(1)}时间内求解。