Introduction to Reconfiguration

Nishimura, Nobuya

doi:10.20944/preprints201709.0055.v1

Cited by 9 publications

(7 citation statements)

References 96 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Other structural considerations of the reconfiguration graph have also been investigated in . Reconfiguration graphs have also been studied for many other decision problems; see for a recent survey.…”

Section: Introductionmentioning

confidence: 99%

Toward Cereceda's conjecture for planar graphs

Eiben

Feghali

2019

Journal of Graph Theory

View full text Add to dashboard Cite

The reconfiguration graph R k ( G ) of the k‐colorings of a graph G has as vertex set the set of all possible k‐colorings of G and two colorings are adjacent if they differ on the color of exactly one vertex. Cereceda conjectured 10 years ago that, for every k‐degenerate graph G on n vertices, R k + 2 ( G ) has diameter scriptO ( n 2 ). The conjecture is wide open, with a best known bound of scriptO ( k n ), even for planar graphs. We improve this bound for planar graphs to 2 scriptO ( n ). Our proof can be transformed into an algorithm that runs in 2 scriptO ( n ) time.

show abstract

Section: Introductionmentioning

confidence: 99%

Toward Cereceda's conjecture for planar graphs

Eiben

Feghali

2019

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…The H-Recolouring problem is part of a growing area known as "combinatorial reconfiguration," a central focus of which is to determine the complexity of deciding whether a given solution to a combinatorial problem can be transformed into another by applying a sequence of allowed modifications. For further background on combinatorial reconfiguration in general, see [1, 8-10, 13, 15-17] and the surveys of van den Heuvel [12], Ito and Suzuki [14] and Nishimura [18].…”

Section: Introductionmentioning

confidence: 99%

Reconfiguring graph homomorphisms on the sphere

Lee

Noel

Siggers

2020

European Journal of Combinatorics

View full text Add to dashboard Cite

Given a loop-free graph H, the reconfiguration problem for homomorphisms to H (also called H-colourings) asks: given two H-colourings f of g of a graph G, is it possible to transform f into g by a sequence of single-vertex colour changes such that every intermediate mapping is an H-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs (e.g. all C4-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever H is a K2,3-free quadrangulation of the 2-sphere (equivalently, the plane) which is not a 4-cycle.If we instead consider graphs G and H with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for H-colourings is PSPACE-complete whenever H is a reflexive K4-free triangulation of the 2-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which H-Recolouring is known to be PSPACE-complete for reflexive instances.

show abstract

“…The fact that there exists an efficient multi-commodity flow for the JS chain can be shown using exactly the same arguments as in Theorem 3.2 in [1]. 12 Without going into all the details, we will give a sketch of this argument. Recall that Sinclair's multi-commodity flow method asks us to define a flow f in the state space graph of the JS chain that routes a fraction π(X)π(Y ) of flow from X to Y for every X, Y ∈ F G .…”

Section: Dense Monotone Graphsmentioning

confidence: 95%

“…The question of irreducibility, as well as being integral to the MCMC method, is studied in its own right under the moniker of reconfiguration problems. Here, one wishes to decide whether the space of solutions to some combinatorial problem is connected (where two solutions are adjacent if one can be obtained from the other by some small prescibed change); see for example the surveys of van den Heuvel [17] and Nishimura [12]. Reconfiguration problems about Hamiltonian cycles have not been widely considered.…”

Section: Related Workmentioning

confidence: 99%

Switch-Based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

Kleer

Patel²,

Stroh³

2020

Electron. J. Combin.

View full text Add to dashboard Cite

We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on $n$ vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least $n/2+7$ can be transformed into each other by switch operations of size at most 10, implying that the switch Markov chain using switches of size at most 10 is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.

show abstract

Introduction to Reconfiguration

Cited by 9 publications

References 96 publications

Toward Cereceda's conjecture for planar graphs

Toward Cereceda's conjecture for planar graphs

Reconfiguring graph homomorphisms on the sphere

Switch-Based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

Contact Info

Product

Resources

About