On the diameter of reconfiguration graphs for vertex colourings

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

doi:10.1016/j.endm.2011.09.028

Cited by 16 publications

(12 citation statements)

References 10 publications

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“…They stated the following conjecture. Bonamy, Johnson, Lignos, Patel, and Paulusma [9] determined sufficient conditions for C k (G) to have a diameter quadratic in the order of G. They showed that k-colourable chordal graphs and chordal bipartite graphs satisfy these conditions and hence have an ℓ-colour diameter that is quadratic in k for ℓ ≥ k + 1 and ℓ = 3, respectively. Bonamy and Bousquet [8] proved a similar result for graphs of bounded treewidth.…”

Section: Question 31 [8]mentioning

confidence: 99%

Reconfiguration of Colourings and Dominating Sets in Graphs

Nasserasr¹

2019

50 Years of Combinatorics, Graph Theory, and Computing

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We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the k-colouring graph C k (G) of a graph G correspond to the proper k-colourings of a graph G, with two k-colourings being adjacent whenever they differ in the colour of exactly one vertex. Similarly, the vertices of the k-edge-colouring graph EC k (G) of g are the proper k-edge-colourings of G, where two k-edge-colourings are adjacent if one can be obtained from the other by switching two colours along an edge-Kempe chain, i.e., a maximal two-coloured alternating path or cycle of edges.The vertices of the k-dominating graph D k (G) are the (not necessarily minimal) dominating sets of G of cardinality k or less, two dominating sets being adjacent in D k (G) if one can be obtained from the other by adding or deleting one vertex. On the other hand, when we restrict the dominating sets to be minimum dominating sets, for example, we obtain different types of domination reconfiguration graphs, depending on whether vertices are exchanged along edges or not.We consider these and related types of colouring and domination reconfiguration graphs. Conjectures, questions and open problems are stated within the relevant sections. * Supported by the Natural Sciences and Engineering Research Council of Canada.1 We use the term connectedness instead of connectivity when referring to the question of whether a graph is connected or not, as the latter term refers to a specific graph parameter.

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Section: Question 31 [8]mentioning

confidence: 99%

Reconfiguration of Colourings and Dominating Sets in Graphs

Nasserasr¹

2019

50 Years of Combinatorics, Graph Theory, and Computing

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“…We first consider the case where a P-node P is in Case (2). Then, there is exactly one pair of a blue token b and a red token r = g * (b), and P = LCA(b, r).…”

Section: Algorithm and Its Correctnessmentioning

confidence: 99%

“…well-known problems, including independent set [12,13,14,16,20], satisfiability [11,18], set cover, clique, matching [14], vertex-coloring [2,3,6], shortest path [15], and so on.…”

Section: Introductionmentioning

confidence: 99%

Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Uehara

2016

WALCOM: Algorithms and Computation

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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. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. The reconfiguration problems tend to be PSPACE-complete in general, and some polynomial time algorithms are shown in restricted cases. Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy for the complexity of finding the shortest sequence has been shown; that is, it is in P, NPcomplete, or PSPACE-complete in certain conditions. In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be NP-complete to find a shortest sequence between them. Namely, finding a shortest sequence between two independent sets can be more difficult than the decision problem of reconfigurability between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs. More precisely, we can find a shortest sequence between two independent sets on a graph G in polynomial time if either G is a proper interval graph, a trivially perfect graph, or a caterpillar. As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token problem for a graph class that requires detours.Recently, the reconfiguration problems attract the attention from the viewpoint of theoretical computer science. The problem arises when we wish to find a stepby-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, that is, an adjacency relation defined on feasible solutions of the original problem. The reconfiguration problems have been studied extensively for several

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“…Since then, reconfiguration versions of various problems have been studied, including maximum independent set, minimum vertex cover, maximum matching, shortest path, graph colorability, and many others [8,20,21,22,26]. Typical questions addressed in these works include the structure or the complexity of determining st-connectivity: whether there is a path from s to t in the reconfiguration graph [8,20,21,22] or connectivity: whether the reconfiguration graph is connected [5,11,17] or upper bounds for the diameter of the reconfiguration graph [6,8,21].…”

Section: Background and Motivationmentioning

confidence: 99%

Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

Mouawad

Nishimura

Pathak

et al. 2015

Automata, Languages, and Programming

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Abstract. Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment s to another satisfying assignment t of a Boolean formula. Earlier work characterized the complexity of finding any (not necessarily the shortest) sequence of flips from one satisfying assignment to another using Schaefer's framework for classification of Boolean formulas. We build on it to provide a trichotomy for the complexity of finding the shortest sequence of flips and show that it is either in P, NP-complete, or PSPACE-complete. Our result adds to the small set of complexity results known for shortest reconfiguration sequence problems by providing an example where the shortest sequence can be found in polynomial time even though its length is not equal to the symmetric difference of the values of the variables in s and t. This is in contrast to all reconfiguration problems studied so far, where polynomial time algorithms for computing the shortest path were known only for cases where the path modified the symmetric difference only.

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On the diameter of reconfiguration graphs for vertex colourings

Cited by 16 publications

References 10 publications

Reconfiguration of Colourings and Dominating Sets in Graphs

Reconfiguration of Colourings and Dominating Sets in Graphs

Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

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