Reconfiguring Independent Sets in Claw-Free Graphs

Bonsma, Paul; Kamiński, Marcin; Wrochna, Marcin

doi:10.1007/978-3-319-08404-6_8

Cited by 60 publications

(72 citation statements)

References 34 publications

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“…• Token Sliding (TS rule) [6,7,10,11,19,23]: This rule corresponds to sliding token, that is, we can slide a single token only along an edge of a graph.…”

Section: Related and Known Resultsmentioning

confidence: 99%

“…• Token Jumping (TJ rule) [7,15,19,23]: A single token can "jump" to any vertex (including non-adjacent one) if it results in an independent set. • Token Addition and Removal (TAR rule) [2,5,13,19,[21][22][23]: We can either add or remove a single token at a time if it results in an independent set of cardinality at least a given threshold minus one.…”

Section: Related and Known Resultsmentioning

confidence: 99%

“…The problem arises when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible and each step abides by a fixed reconfiguration rule (i.e., an adjacency relation defined on feasible solutions of the original problem). This kind of reconfiguration problem has been studied extensively for several well-known problems, including independent set [2,5,7,10,11,13,15,19,[21][22][23], satisfiability [9,20], set cover, clique, matching [13], vertexcoloring [3,6,8,23], list edge-coloring [14,17], list L(2, 1)-labeling [16], subset sum [12], shortest path [4,18], and so on.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial-Time Algorithm for Sliding Tokens on Trees

Demaine¹,

Demaine²,

Fox-Epstein

et al. 2014

Algorithms and Computation

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Abstract. Suppose that we are given two independent sets I b and Ir of a graph such that |I b | = |Ir|, and imagine that a token is placed on each vertex in I b . Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I b into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between I b and Ir whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

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“…• Token Sliding (TS rule) [6,7,10,11,19,23]: This rule corresponds to sliding token, that is, we can slide a single token only along an edge of a graph.…”

Section: Related and Known Resultsmentioning

confidence: 99%

Section: Related and Known Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polynomial-Time Algorithm for Sliding Tokens on Trees

Demaine¹,

Demaine²,

Fox-Epstein

et al. 2014

Algorithms and Computation

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“…Polynomial-time algorithms have been developed for shortest transformation under TS for proper interval graphs, trivially perfect graphs, and caterpillars [58] (though NP-hard in general for all reconfiguration steps [17]), for connectivity under TS for interval graphs [56] and under TAR for cographs [55], and for finding an actual reconfiguration sequence under TS for trees (as an extension of the reachability algorithm) [59]. It has also been shown that there is an infinite family of instances on paths for which the length of the reconfiguration sequence is at least quadratic in the number of vertices [59]; in contrast, for every yes-instance of reachability on cographs, there is a bound on the length of the reconfiguration sequence [42] as well as a bound on the diameter for claw-free graphs [57]. In other work, the idea of reconfiguration of independent sets in graphs has been extended to reconfiguration of independent sets in matroids [64].…”

Section: Analysis Using Graph Classes For Independent Setmentioning

confidence: 99%

Introduction to Reconfiguration

Nishimura¹

2017

Preprint

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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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“…Reconfiguration graphs have been studied for a number of combinatorial problems; the questions asked are typically (as we have seen for colouring) is the graph connected?, what is the diameter of the graph (or of its connected components)?, how difficult is it to decide whether there is a path between a pair of given solutions? Problems studied include boolean satisfiability [13,21],clique and vertex cover [16], independent set [6,20], list edge colouring [17],shortest path [3,4], and subset sum [15] (see also a recent survey [14]). Recent work has included looking at finding the shortest path in the reconfiguration graph between given solutions [19], and studying the fixed-parameter-tractability of these problems [7,18,23,24].…”

Section: (The Answer Is Always Yes)mentioning

confidence: 99%

A Reconfigurations Analogue of Brooks’ Theorem

Feghali

Johnson

Paulusma

2014

Mathematical Foundations of Computer Science 2014

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let G be a simple undirected graph on n vertices with maximum degree ∆. Brooks' Theorem states that G has a ∆-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > ∆, a ∆-colouring of G can be obtained by a sequence of O(n 2 ) recolourings using only the original k colours unless -G is a complete graph or a cycle with an odd number of vertices, or -k = ∆ + 1, G is ∆-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the kcolourings of G. The vertex set of R k (G) is the set of all possible kcolourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that -if k ≤ ∆(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, -if k ≥ ∆(G) + 2, then R k (G) is connected and has diameter O(n 2 ). We complete this structural classification by settling the missing case:-if k = ∆(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2 ). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree ∆ belong to the same component of R k (G) by settling the case k = ∆(G) + 1. The problem is -O(n 2 ) time solvable for k = 3, -PSPACE-complete for 4 ≤ k ≤ ∆(G), -O(n) time solvable for k = ∆(G) + 1, -O(1) time solvable for k ≥ ∆(G) + 2 (the answer is always yes).

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Reconfiguring Independent Sets in Claw-Free Graphs

Cited by 60 publications

References 34 publications

Polynomial-Time Algorithm for Sliding Tokens on Trees

Polynomial-Time Algorithm for Sliding Tokens on Trees

Introduction to Reconfiguration

A Reconfigurations Analogue of Brooks’ Theorem

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