Finding paths between 3-colorings

Cereceda, Luis; Heuvel, Jan van den; Johnson, Matthew

doi:10.1002/jgt.20514

Cited by 137 publications

(173 citation statements)

References 9 publications

(7 reference statements)

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“…Note that k-dominating and k-independent graph are similar to recent work in graph colouring, Authors in [6,7,8,9] studied the connectedness of k-colouring graphs. Also they studied their hamiltonicity.…”

Section: Introductionmentioning

confidence: 51%

THE k-INDEPENDENT GRAPH OF A GRAPH

Fatehi¹,

Alikhani²,

Khalaf³

2017

AADM

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Let G = (V, E) be a simple graph. A set I ⊆ V is an independent set, if no two of its members are adjacent in G. The k-independent graph of G, I k (G), is defined to be the graph whose vertices correspond to the independent sets of G that have cardinality at most k. Two vertices in I k (G) are adjacent if and only if the corresponding independent sets of G differ by either adding or deleting a single vertex. In this paper, we obtain some properties of I k (G) and compute it for some graphs.Mathematics Subject Classification: 05C60, 05C69.

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Section: Introductionmentioning

confidence: 51%

THE k-INDEPENDENT GRAPH OF A GRAPH

Fatehi¹,

Alikhani²,

Khalaf³

2017

AADM

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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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“…The problem of deciding whether two 3-colourings of a graph G are in the same component of R 3 (G) was shown to be solvable in time O(n 2 ) in [12]; it was also proved that the diameter of any component of R 3 (G) is O(n 2 ). In contrast, in [5] the analagous problem for k-colourings, k ≥ 4, was shown to be PSPACE-complete, and examples of reconfiguration graphs with components of superpolynomial diameter were given.…”

Section: Introductionmentioning

confidence: 99%

A Reconfigurations Analogue of Brooks’ Theorem

Feghali

Johnson

Paulusma

2014

Lecture Notes in Computer Science

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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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Finding paths between 3-colorings

Cited by 137 publications

References 9 publications

THE k-INDEPENDENT GRAPH OF A GRAPH

THE k-INDEPENDENT GRAPH OF A GRAPH

Polynomial-Time Algorithm for Sliding Tokens on Trees

A Reconfigurations Analogue of Brooks’ Theorem

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