The Coloring Reconfiguration Problem on Specific Graph Classes

Hatanaka, Tatsuhiko; Ito, Takehiro; Zhou, Xiao

doi:10.1007/978-3-319-71150-8_15

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…Another well-studied spacial case is CR [1,2,4,5,6,7,13,14,15,18,19,21,23,29,34,35,41]. This problem is PSPACE-complete for k ≥ 4 and bipartite planar graphs [4] but in P for k ≤ 3 [15].…”

Section: Known and Related Resultsmentioning

confidence: 99%

“…We note that these results (including tractability one) can be extended for LCR. Moreover, it is known that the problem remains PSPACE-complete even if k is a fixed constant for several graph classes such as line graphs (for any fixed k ≥ 5) [35], bounded bandwidth graphs [41], and chordal graphs [23]. On the other hand, several polynomial-time algorithms are known for subclasses of chordal graphs such as trivially perfect graphs, split graphs [23], and (k − 2)-connected chordal graphs [6].…”

Section: Known and Related Resultsmentioning

confidence: 99%

“…As an example, we consider the (vertex) coloring reconfiguration problem [4,6,15,23,41], which is one of the most well-studied reconfiguration problems. Let G = (V, E) be a graph and let D be a set of k colors.…”

Section: Introductionmentioning

confidence: 99%

“…The reconfiguration problem for CSP has been studied as several spacial cases such as Boolean constraint satisfiability reconfiguration [5,12,20,30,31,32,38], homomorphism reconfiguration [8,9,10,11,40,41], and (list) coloring reconfiguration [1,2,4,5,6,7,13,14,15,18,19,21,22,23,24,27,29,34,35,41]. We will briefly summarize these results in the next section.…”

Section: Introductionmentioning

confidence: 99%

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Complexity of Reconfiguration Problems for Constraint Satisfaction

Hatanaka¹,

Ito²,

Zhou³

2018

Preprint

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Constraint satisfaction problem (CSP) is a well-studied combinatorial search problem, in which we are asked to find an assignment of values to given variables so as to satisfy all of given constraints. We study a reconfiguration variant of CSP, in which we are given an instance of CSP together with its two satisfying assignments, and asked to determine whether one assignment can be transformed into the other by changing a single variable assignment at a time, while always remaining satisfying assignment. This problem generalizes several well-studied reconfiguration problems such as Boolean satisfiability reconfiguration, vertex coloring reconfiguration, homomorphism reconfiguration. In this paper, we study the problem from the viewpoints of polynomial-time solvability and parameterized complexity, and give several interesting boundaries of tractable and intractable cases.

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Section: Known and Related Resultsmentioning

confidence: 99%

Section: Known and Related Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Complexity of Reconfiguration Problems for Constraint Satisfaction

Hatanaka¹,

Ito²,

Zhou³

2018

Preprint

Self Cite

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“…In this paper, we study the reconfiguration problem for (vertex) colorings in a graph, called the coloring reconfiguration problem, which was introduced by Bonsma and Cereceda [3]. Manuscript * A preliminary version of this paper has been presented at the 11th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2017) [11]. This work is partially supported by JST CREST Grant Number JPMJCR1402, and by JSPS KAKENHI Grant Numbers JP16J02175, JP16K00003, and JP16K00004, Japan.…”

Section: Our Problemmentioning

confidence: 99%

The Coloring Reconfiguration Problem on Specific Graph Classes

Hatanaka

Ito

Zhou

2019

IEICE Trans. Inf. & Syst.

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We study the problem of transforming one (vertex) ccoloring of a graph into another one by changing only one vertex color assignment at a time, while at all times maintaining a c-coloring, where c denotes the number of colors. This decision problem is known to be PSPACE-complete even for bipartite graphs and any fixed constant c ≥ 4. In this paper, we study the problem from the viewpoint of graph classes. We first show that the problem remains PSPACE-complete for chordal graphs even if c is a fixed constant. We then demonstrate that, even when c is a part of input, the problem is solvable in polynomial time for several graph classes, such as k-trees with any integer k ≥ 1, split graphs, and trivially perfect graphs.

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Characterizing circular colouring mixing for pq<4 $\frac{p}{q}\lt 4$

Brewster

Moore

2022

Journal of Graph Theory

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Given a graph G $G$, the k $k$‐mixing problem asks: Starting with a k $k$‐colouring of G $G$, can one obtain all k $k$‐colourings of G $G$ by changing the colour of only one vertex at a time, while at each step maintaining a k $k$‐colouring? More generally, for a graph H $H$, the H $H$‐mixing problem asks: Can one obtain all homomorphisms G → H $G\to H$, starting from one homomorphism f $f$, by changing the image of only one vertex at a time, while at each step maintaining a homomorphism G → H $G\to H$? This paper focuses on a generalization of k $k$‐colourings, namely, (p , q ) $(p,q)$‐circular colourings. We show that when 2 < p q < 4 $2\lt \frac{p}{q}\lt 4$, a graph G $G$ is (p , q ) $(p,q)$‐mixing if and only if for any (p , q ) $(p,q)$‐colouring f $f$ of G $G$, and any cycle C $C$ of G $G$, the wind of the cycle under the colouring equals a particular value (which intuitively corresponds to having no wind). As a consequence we show that (p , q ) $(p,q)$‐mixing is closed under a restricted homomorphism called a fold. Using this, we deduce that (2 k + 1 , k ) $(2k+1,k)$‐mixing is co‐NP‐complete for all k ∈ double-struckN $k\in {\mathbb{N}}$, and by similar ideas we show that if the circular chromatic number of a connected graph G $G$ is 2 k + 1 k $\frac{2k+1}{k}$, then G $G$ folds to C2 k + 1 ${C}_{2k+1}$. We use the characterization to settle a conjecture of Brewster and Noel, specifically that the circular mixing number of bipartite graphs is 2. Lastly, we give a polynomial time algorithm for (p , q ) $(p,q)$‐mixing in planar graphs when 3 ≤ p q < 4 $3\le \frac{p}{q}\lt 4$.

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The Coloring Reconfiguration Problem on Specific Graph Classes

Cited by 4 publications

References 13 publications

Complexity of Reconfiguration Problems for Constraint Satisfaction

Complexity of Reconfiguration Problems for Constraint Satisfaction

The Coloring Reconfiguration Problem on Specific Graph Classes

Characterizing circular colouring mixing for pq<4 $\frac{p}{q}\lt 4$

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