Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

Bonamy, Marthe; Johnson, Matthew; Lignos, Ioannis; Patel, Viresh; Paulusma, Daniël

doi:10.1007/s10878-012-9490-y

Cited by 79 publications

(137 citation statements)

References 17 publications

(18 reference statements)

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“…(We simply say a reconfiguration problem on a search problem A if the reconfiguration problem considers feasible solutions of A as the intermediate solutions.) This reconfiguration framework has been applied to several well-known problems, including independent set [9], [10], [12], [14], satisfiability [8], clique, matching [10], vertex-coloring [2], [4], etc.…”

Section: Related and Known Resultsmentioning

confidence: 99%

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Reconfiguration of Vertex Covers in a Graph

Ito

Hiroyuki

Zhou

2016

IEICE Trans. Inf. & Syst.

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SUMMARY Suppose that we are given two vertex covers C 0 and C t of a graph G, together with an integer threshold k ≥ max{|C 0 |, |C t |}. Then, the vertex cover reconfiguration problem is to determine whether there exists a sequence of vertex covers of G which transforms C 0 into C t such that each vertex cover in the sequence is of cardinality at most k and is obtained from the previous one by either adding or deleting exactly one vertex. This problem is PSPACE-complete even for planar graphs. In this paper, we first give a linear-time algorithm to solve the problem for even-hole-free graphs, which include several well-known graphs, such as trees, interval graphs and chordal graphs. We then give an upper bound on k for which any pair of vertex covers in a graph G has a desired sequence. Our upper bound is best possible in some sense.

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Section: Related and Known Resultsmentioning

confidence: 99%

“…Of course, we wish to keep monitoring all links even during the transformation. This situation can be formulated by the concept of reconfiguration problems that have been extensively studied in recent literature [2], [4], [8]- [10], [12], [14].…”

Section: Introductionmentioning

confidence: 99%

Reconfiguration of Vertex Covers in a Graph

Ito

Hiroyuki

Zhou

2016

IEICE Trans. Inf. & Syst.

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“…It has been shown that for every graph, the reconfiguration graph is connected when k ≥ ∆(G) + 2 [82], and in fact for k ≥ col(G) + 2 [83]. Investigations into c-color-dense graphs proved that the reconfiguration graph is connected for chordal and chordal bipartite graphs [84], subsequently generalized to hold for any k ≥ tw(G) + 2 [85] and k ≥ χ g (G) + 1 [85].…”

Section: K-coloring Reconfigurationmentioning

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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“…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%

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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Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

Cited by 79 publications

References 17 publications

Reconfiguration of Vertex Covers in a Graph

Reconfiguration of Vertex Covers in a Graph

Introduction to Reconfiguration

Polynomial-Time Algorithm for Sliding Tokens on Trees

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