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A graph vertex-subset problem defines which subsets of the vertices of aninput graph are feasible solutions. We view a feasible solution as a set oftokens placed on the vertices of the graph. A reconfiguration variant of avertex-subset problem asks, given two feasible solutions of size $k$, whetherit is possible to transform one into the other by a sequence of token slides(along edges of the graph) or token jumps (between arbitrary vertices of thegraph) such that each intermediate set remains a feasible solution of size $k$.Many algorithmic questions present themselves in the form of reconfigurationproblems: Given the description of an initial system state and the descriptionof a target state, is it possible to transform the system from its initialstate into the target one while preserving certain properties of the system inthe process? Such questions have received a substantial amount of attentionunder the so-called combinatorial reconfiguration framework. We considerreconfiguration variants of three fundamental underlying graph vertex-subsetproblems, namely Independent Set, Dominating Set, and Connected Dominating Set.We survey both older and more recent work on the parameterized complexity ofall three problems when parameterized by the number of tokens $k$. The emphasiswill be on positive results and the most common techniques for the design offixed-parameter tractable algorithms.

A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems