The complexity of dominating set reconfiguration

Haddadan, Arash; Ito, Takehiro; Mouawad, Amer E.; Nishimura, Nobuya; Ono, Hiroyuki; Suzuki, Ai; Tebbal, Youcef

doi:10.1016/j.tcs.2016.08.016

Cited by 32 publications

(26 citation statements)

References 13 publications

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“…Wrochna [47] proved the PSPACE-completeness of the reachability problem, where the reconfiguration step is changing a single symbol, based on the classic proof of undecidability for general Thue systems [48]. Using this result, the following reachability problems were shown to be PSPACE-complete for graphs of bounded bandwidth: SHORTEST PATH [47], k-COLORING [47], LIST COLORING [47], INDEPENDENT SET [47], VERTEX COVER [49], FEEDBACK VERTEX SET [49], ODD CYCLE TRANSVERSAL [49], INDUCED FOREST [49], INDUCED BIPARTITE SUBGRAPH [49], and DOMINATING SET [50]. In investigations of the parameterized complexity (defined in Section 5) of NCL under the parameters treewidth, maximum degree, and length of the reconfiguration sequence, H-WORD RECONFIGURATION has been used to strengthen the framework to remain PSPACE-complete for graph of bounded bandwidth [51].…”

Section: Tools For Proving the Complexity Of Reachabilitymentioning

confidence: 99%

“…Subsequent (and ongoing) research has focused on the question of the boundaries between tractability and intractability for the source problem based on various graph classes, both to determine whether the boundaries are the same for the source problem and for reachability, and to determine whether the boundaries for reachability are the same for both TS and TAR/TJ. Analyses of this type have also been considered for other problems for which reachability is PSPACE-complete on general graphs, including CLIQUE RECONFIGURATION [53], CLUSTER VERTEX DELETION RECONFIGURATION [53], DOMINATING SET RECONFIGURATION [50], and VERTEX COVER RECONFIGURATION (discussed in more detail in Section 5). Figure 5 and Table 1 represent the results known so far.…”

Section: Analysis Using Graph Classes For Independent Setmentioning

confidence: 99%

“…Under TAR, the reachability problem is PSPACE-complete even for graphs of bounded bandwidth, split graphs, planar graphs, and bipartite graphs [50], whereas linear-time algorithms have been developed for cographs, trees, and interval graphs [50]. Although W[1]-hard when parameterized by k, the problem is fixed-parameter tractable when parameterized by k when the input graph does not contain large bicliques (including bounded degeneracy and nowhere dense graphs) [74].…”

Section: Other Structural Problemsmentioning

confidence: 99%

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Introduction to Reconfiguration

2018

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

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Section: Tools For Proving the Complexity Of Reachabilitymentioning

confidence: 99%

Section: Analysis Using Graph Classes For Independent Setmentioning

confidence: 99%

Section: Other Structural Problemsmentioning

confidence: 99%

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Introduction to Reconfiguration

2018

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“…The reconfiguration problems have attracted much attention recently because of their applications as well as theoretical interest. See, for example, a survey [1] and references of [2,3].…”

Section: Introductionmentioning

confidence: 99%

Complexity of Hamiltonian Cycle Reconfiguration

Takaoka

2018

Algorithms

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The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time.

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“…in the matching. This kind of reconfiguration problems has been studied extensively for several wellknown combinatorial problems, including satisfiability [6,17,19], independent set [4,5,15], vertex cover [13,20], clique [14], dominating set [7,8], vertex-coloring [1,3,9], and so on. (See also a survey [10].…”

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Reconfiguration of maximum-weight b-matchings in a graph

et al. 2018

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Consider a graph such that each vertex has a nonnegative integer capacity and each edge has a positive integer weight. Then, a b-matching in the graph is a multi-set of edges (represented by an integer vector on edges) such that the total number of edges incident to each vertex is at most the capacity of the vertex. In this paper, we study a reconfiguration variant for maximum-weight b-matchings: For two given maximum-weight b-matchings in a graph, we are asked to determine whether there exists a sequence of maximum-weight b-matchings in the graph between them, with subsequent b-matchings obtained by removing one edge and adding another. We show that this reconfiguration problem is solvable in polynomial time for instances with no integrality gap. Such instances include bipartite graphs with any capacity function on vertices, and 2-matchings in general graphs. Thus, our result implies that the reconfiguration problem for maximum-weight matchings can be solved in polynomial time for bipartite graphs.

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The complexity of dominating set reconfiguration

Cited by 32 publications

References 13 publications

Introduction to Reconfiguration

Introduction to Reconfiguration

Complexity of Hamiltonian Cycle Reconfiguration

Reconfiguration of maximum-weight b-matchings in a graph

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