maximum cardinality of a minimal dominating set of G.
Very recently, Suzuki, Mouawad, and Nishimura [74] extended the results for the connectivity of R(G) under the TAR rule showing that R(G) is connected for k = n − m, when G has at least m + 1 independent edges. They also give counterexamples and thus giving an answer to a question in [36] on whether R(G) is connected for any graph, when k = Γ(G) + 1. The examples are planar, multi-partite and of bounded treewidth graphs. Finally, they demonstrate an infinite family of graphs of exponential diameter, when k = γ(G) + 1, which is the minimum value for k for the TAR model – if k = γ(G), then we cannot delete vertices, but only swap, which is possible only under the TS and TJ rules.
3.2.6 Problems Remaining in P
Shortest Path for general instances, 4-Colouring for bipartite and planar graphs and SAT for tight relations are all polynomially solvable, but their reconfiguration version is PSPACE-complete, as seen in Sections 3.2.2, 2.4.2 and 3.1.1, respectively.
Perhaps the result on Shortest Path was the most surprising of all, as it is the only known recon-figuration problem which is PSPACE-complete on general instances while the original version is in P. For example, Minimum Spanning Tree Reconfiguration and Matching Reconfiguration are in P both originally and as a reconfiguration problem [44]. Actually, the authors prove that Matroid Reconfiguration is in P, which generalises the result for Minimum Spanning Tree.
3.3 Parameterized Complexity and Reconfiguration
Lately, there has been an increasing interest to examine the tractability of reconfiguration prob-lems through a different perspective. Since a lot of probprob-lems are PSPACE-complete, it seems reasonable to look at their parameterized complexity [24] or how useful it is to approximate their solutions.
3.3.1 Classes
Some problems accept an algorithm which requires time polynomial on the input size (e.g. the number of vertices), but can be exponential for some parameter k of the problem. If this parameter can be fixed, i.e. its size does not depend on the input size n of the problem, then the problem belongs in the complexity class FPT (Fixed Parameter Tractable) or we say that the problem is FPT. Other problems remain intractable even when one or more of their parameters are fixed.
Those problems belong to the W[t], t = 1, 2, ... complexity classes, with FPT = W[0]. These classes form the W-hierarchy and they are such that W[i] ⊆ W[j], for all 1 ≤ i ≤ j. If a problem is W[i]-complete, then there is an FPT-time reduction to other problems which are W[i]-complete.
For example, a problem is W[1]-complete if it can reduce to CLIQUE or INDEPENDENT SET in FPT-time and a problem is W[2]-complete if it can reduce to DOMINATING SET using an FPT algorithm. The W[i] hierarchy can also be formally defined in relation to combinatorial circuits of weft i. For more details and precise definitions the reader should refer to one of the parameterized complexity textbooks available, for example, see [24].
3.3.2 Bounding Solutions and Reconfiguration Sequences
Mouawad et al. [65] first suggested two straightforward parameterizations of reconfiguration prob-lems; to bound the number of solutions k and/or the length ` of the reconfiguration sequence between two solutions in R(G). They adapt or extend methods used in the area of parameterized complexity in order to obtain polynomial reconfiguration kernels in bounding the number of solu-tions, and they manage to do this for the Feedback Vertex Set Reconfiguration and Bounded Hitting Set Reconfiguration problems. On the contrary, they show that Unbounded Hitting Set Reconfigu-ration and Dominating Set ReconfiguReconfigu-ration are W[2]-hard, when parameterized by k + `.
Independent Set, Vertex Cover, Dominating Set, and Graph Colouring Reconfiguration
Mouawad et al. [65] also give a general approach on reconfiguration versions of problems with hereditary properties, classifying them as W[1]-hard, for example Independent Set parameterized
3.3. PARAMETERIZED COMPLEXITY AND RECONFIGURATION 33 by k+` and Vertex Cover parameterized by `. They also show that Dominating Set Reconfiguration parameterised by k + ` is W[2]-hard, where ` is an upper bound on the length of the reconfiguration sequence.
For the latter, there has been more work disseminated very recently, aiming to find restricted instances for which the two problems become fixed-parameter tractable (FPT). Mouawad et al. [64]
show that Vertex Cover Reconfiguration remains W[1]-hard for bipartite graphs, which is important in the sense that the original problem is in P for the same class, and FPT for graphs of bounded degree. And finally for the Independent Set Reconfiguration, also very recently, Ito et al. [47] show that the problem under the TJ rule is W[1]-hard, when parameterised by the size of the independent sets, but FPT, when parameterized by both the size of the independent sets and the maximum degree. Even more recently [66] both problems were shown to be FPT for planar graphs.
Finally, Johnson et al. [52] and also Bonsma and Mouawad [12] independently showed that k-Colouring Reconfiguration is FPT for k ≥ 3, when parameterized by the length of the reconfig-uration sequence.
Reconfiguration on Graphs of Bounded Tree-width, Band-width and Tree-depth
Mouawad et al. [66] examine several reconfiguration problems for graphs of bounded tree-width t and they prove that most of them remain PSPACE-complete: e.g. Independent Set, Vertex Cover, Feedback Vertex Set. However, they also show that they are FPT, when parameterized by t. They manage to show this by introducing a technique which defines reconfiguration problems in monadic second order logic.
Wrochna [78] show that k-Colouring, Independent Set, and Shortest Path reconfiguration prob-lems remain PSPACE-complete even for graphs of bounded bandwidth, which restricts instances of the problem more than tree-width and path-width do.