Connectedness of the graph of vertex-colourings

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

doi:10.1016/j.disc.2007.07.028

Cited by 116 publications

(197 citation statements)

References 7 publications

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“…Moreover, the study of reconfiguration yields insights into the structure of the solution space of the underlying problem, crucial for the design of efficient algorithms. In fact, one of the initial motivations behind such questions was to study the performance of heuristics [9] and random sampling methods [4], where connectivity and other properties of the solution space play a crucial role. Even though reconfiguration gained popularity in the last decade or so, the notion of exploring the solution space of a given problem has been previously considered in numerous settings.…”

Section: Introductionmentioning

confidence: 99%

Reconfiguration of dominating sets

2015

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Abstract. We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of D k (G), the graph consisting of a vertex for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that D Γ (G)+1 (G) is not necessarily connected, for Γ (G) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for b ≥ 3. Moreover, we construct an infinite family of graphs such that D γ(G)+1 (G) has exponential diameter, for γ(G) the minimum size of a dominating set. On the positive side, we show that Dn−m(G) is connected and of linear diameter for any graph G on n vertices having at least m + 1 independent edges.

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Section: Introductionmentioning

confidence: 99%

Reconfiguration of dominating sets

2015

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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: 49%

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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“…The study of reconfiguration graphs of colourings began in [10,11]. 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 ).…”

Section: Introductionmentioning

confidence: 99%

“…It is possible that the number of isolated vertices is zero (that is, there are no frozen (∆ + 1)-colourings; for example, consider 4-colourings of K 3,3 ), or that there are only isolated vertices (consider R 4 (K 4 ) for instance; and Brooks' theorem tells us that complete graphs are the only graphs for which R ∆+1 (G) is edgeless since other graphs have colourings in which only ∆ colours are used and by recolouring any vertex with the unused colour we find a neighbouring colouring). We observe that the requirement that ∆ ≥ 3 is necessary since, for example R 3 (C n ), n odd, has more than one component [10,11].…”

Section: Introductionmentioning

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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Connectedness of the graph of vertex-colourings

Cited by 116 publications

References 7 publications

Reconfiguration of dominating sets

Reconfiguration of dominating sets

THE k-INDEPENDENT GRAPH OF A GRAPH

A Reconfigurations Analogue of Brooks’ Theorem

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