Parameterized complexity of dominating set variants in almost cluster and split graphs

We consider structural parameterizations of several variants of Dominating Set in the parameter ecology program. We give improved FPT algorithms and lower bounds under well-known conjectures for Dominating Set and its variants 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). For example, we show that when parameterized by the deletion distance k to cluster graphs: Dominating Set, Independent Dominating Set, Dominating Clique, Efficient Dominating Set and Total Dominating Set can be solved in $3^k{n}^{O\left(1\right)}$-time. Additionally, when parameterized by the deletion distance k to split graphs, we prove that Efficient Dominating Set can be solved in $3^{k/2}{n}^{O\left(1\right)}$-time breaking the $2^k$ barrier.