Token Sliding on Split Graphs

Belmonte, Rémy; Kim, Eun Jung; Lampis, Michael; Mitsou, Valia; Otachi, Yota; Sikora, Florian

doi:10.1007/s00224-020-09967-8

Cited by 30 publications

(25 citation statements)

References 19 publications

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“…Lokshtanov and Mouawad [25] showed that, in bipartite graphs, Token Jumping is NP-complete while Token Sliding remains PSPACE-complete. In split graphs, Token Jumping is a trivial problem while Token Sliding is PSPACE-complete [2]. In addition to the classes above, Token Jumping can be decided in polynomial time for even-hole-free graphs [21].…”

Section: :3mentioning

confidence: 99%

On Girth and the Parameterized Complexity of Token Sliding and Token Jumping

et al. 2021

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In the Token Jumping problem we are given a graph G = (V, E) and two independent sets S and T of G, each of size k ≥ 1. The goal is to determine whether there exists a sequence of k-sized independent sets in G, S0, S1, . . . , S , such that for every i, |Si| = k, Si is an independent set, S = S0, S = T , and |Si∆Si+1| = 2. In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of G, then the problem asks for a sequence of independent sets which transforms S to T by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on C4-free bipartite graphs when parameterized by k. For Token Jumping, we in fact show that the problem admits a polynomial kernel on {C3, C4}-free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant p ≥ 4, both problems are W[1]-hard on {C4, . . . , Cp}-free graphs and Token Sliding remains W[1]-hard even on bipartite graphs. ACM Subject ClassificationMathematics of computing → Paths and connectivity problems; Mathematics of computing → Graph algorithms; Mathematics of computing → Combinatoric problems

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Section: :3mentioning

confidence: 99%

On Girth and the Parameterized Complexity of Token Sliding and Token Jumping

et al. 2021

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“…Even on bipartite graphs, VCR remains PSPACE-complete under TS, and NP-complete under each of TJ and TAR [20]. On chordal graphs, VCR is known to be PSPACE-complete under TS [4]. On the positive side, polynomialtime algorithms have been designed for VCR on even-hole-free graphs (and therefore chordal graphs) under each of TJ and TAR [18], on bipartite permutation graphs and bipartite distance-hereditary graphs [12] under TS, on cographs [6,18], claw-free graphs [7], interval graphs [5,18], and trees [10,18] under each of TS, TJ, and TAR.…”

Section: Introductionmentioning

confidence: 99%

Reconfiguring k-path Vertex Covers

Hoang

Suzuki

Yagita

2020

WALCOM: Algorithms and Computation

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A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The k-Path Vertex Cover Reconfiguration (k-PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of k-PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: TS, TJ, and TAR. The problem for k = 2, known as the Vertex Cover Reconfiguration (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for k-PVCR. In particular, we prove a complexity dichotomy for k-PVCR on general graphs: on those whose maximum degree is 3 (and even planar), the problem is PSPACE-complete, while on those whose maximum degree is 2 (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for k-PVCR on trees under each of TJ and TAR. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

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“…Our goal is to find a sequence of independent sets that represents a step-by-step modification from one of the given independent sets to the other. There are three local modification rules studied in the literature: Token Addition and Removal (TAR) [3,19,22], Token Jumping (TJ) [4,5,16,17,18], and Token Sliding (TS) [1,2,8,10,13,15,21]. Under TAR, given a threshold k, we can remove or add any vertices as long as the resultant independent set has size at least k. (When we want to specify the threshold k, we call the rule TAR(k).)…”

Section: Introductionmentioning

confidence: 99%

“…For claw-free graphs, the problem is solvable in polynomial time under all three rules [4]. For even-hole-free graphs (graphs without induced cycles of even length), the problem is known to be polynomial-time solvable under TAR and TJ [19], while it is PSPACE-complete under TS even for split graphs [1]. Under TS, forests [8] and interval graphs [2] form maximal known subclasses of even-hole-free graphs for which Independent Set Reconfiguration is polynomial-time solvable.…”

Section: Introductionmentioning

confidence: 99%

Independent Set Reconfiguration Parameterized by Modular-Width

et al. 2020

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Independent Set Reconfiguration is one of the most well-studied problems in the setting of combinatorial reconfiguration. It is known that the problem is PSPACE-complete even for graphs of bounded bandwidth. This fact rules out the tractability of parameterizations by most well-studied structural parameters as most of them generalize bandwidth. In this paper, we study the parameterization by modular-width, which is not comparable with bandwidth. We show that the problem parameterized by modular-width is fixed-parameter tractable under all previously studied rules TAR, TJ, and TS. The result under TAR resolves an open problem posed by Bonsma [WG 2014, JGT 2016.

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Token Sliding on Split Graphs

Cited by 30 publications

References 19 publications

On Girth and the Parameterized Complexity of Token Sliding and Token Jumping

On Girth and the Parameterized Complexity of Token Sliding and Token Jumping

Reconfiguring k-path Vertex Covers

Independent Set Reconfiguration Parameterized by Modular-Width

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