Abstract
In the Vertex Cover Reconfiguration (VCR) problem, given a graph G, positive integers k and ℓ and two vertex covers S and T of G of size at most k, we determine whether S can be transformed into T by a sequence of at most ℓ vertex additions or removals such that every operation results in a vertex cover of size at most k. Motivated by results establishing the W [ 1 ] -hardness of VCR when parameterized by ℓ, we delineate the complexity of the problem restricted to various graph classes. In particular, we show that VCR remains W [ 1 ] -hard on bipartite graphs, is NP -hard, but fixed-parameter tractable on (regular) graphs of bounded degree and more generally on nowhere dense graphs and is solvable in polynomial time on trees and (with some additional restrictions) on cactus graphs.
Highlights
Under the reconfiguration framework, we consider structural and algorithmic questions related to the solution space of a search problem Q
In earlier work establishing the W[1]-hardness of the Vertex Cover Reconfiguration (VCR) problem parameterized byon general graphs, it was shown that the problem becomes fixed-parameter tractable whenever = |SR ∪ TA | [10]
Since the vertex cover problem is known to be solvable in time polynomial in n on bipartite graphs, our result is, to the best of our knowledge, the first example of a problem solvable in polynomial time whose reconfiguration version is W[1]-hard
Summary
We consider structural and algorithmic questions related to the solution space of a search problem Q. The Vertex Cover Reconfiguration (VCR) problem was shown to be fixed-parameter tractable when parameterized by k and W[1]-hard when parameterized by [10]; in RVC ( G, 0, k), each feasible solution for instance G is a vertex cover of size at most k (a subset S ⊆ V ( G ) such that each edge of the graph has at least one endpoint in S) and two solutions are adjacent if one can be obtained from the other by the addition or removal of a single vertex of G Motivated by these results, we embark on a systematic investigation of the parameterized complexity of the problem restricted to various graph classes. We show using completely different techniques, and at the cost of a much worse running time, that VCR, as well as a host of other reconfiguration problems are fixed-parameter tractable on nowhere dense classes of graphs
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