  A graph vertex-subset problem defines which subsets of the vertices of an
input graph are feasible solutions. We view a feasible solution as a set of
tokens placed on the vertices of the graph. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions of size $k$, whether
it 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 the
graph) such that each intermediate set remains a feasible solution of size $k$.
Many algorithmic questions present themselves in the form of reconfiguration
problems: Given the description of an initial system state and the description
of a target state, is it possible to transform the system from its initial
state into the target one while preserving certain properties of the system in
the process? Such questions have received a substantial amount of attention
under the so-called combinatorial reconfiguration framework. We consider
reconfiguration variants of three fundamental underlying graph vertex-subset
problems, namely Independent Set, Dominating Set, and Connected Dominating Set.
We survey both older and more recent work on the parameterized complexity of
all three problems when parameterized by the number of tokens $k$. The emphasis
will be on positive results and the most common techniques for the design of
fixed-parameter tractable algorithms.

Computational Complexity;Discrete Mathematics;Data Structures and Algorithms;Combinatorics