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Finding Shortest Paths between Graph Colourings
[article]

2014
*
arXiv
*
pre-print

The k-

arXiv:1403.6347v4
fatcat:uibgan3lirbkpabxz2h5x6pely
*colouring*reconfiguration problem asks whether, for a given*graph*G, two proper k-*colourings*α and β of G, and a positive integer ℓ, there exists a sequence of at most ℓ+1 proper k-*colourings*of ... G which starts with α and ends with β and where successive*colourings*in the sequence differ on exactly one vertex of G. ... . ⊓ ⊔ Recolouring: Changing the Height of a Vertex In order to understand the role of vertex heights in*finding**shortest**paths**between**colourings*, we investigate how the heights of vertices change along ...##
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Finding Shortest Paths Between Graph Colourings
[chapter]

2014
*
Lecture Notes in Computer Science
*

2016) '

doi:10.1007/978-3-319-13524-3_19
fatcat:ll6gmmverzdqtovuff65sxinnu
*Finding**shortest**paths**between**graph**colourings*.', Algorithmica., 75 (2). pp. 295-321. Further information on publisher's website: http://dx. ... The k-*colouring*reconfiguration problem asks whether, for a given*graph*G, two proper k-*colourings*α and β of G, and a positive integer , there exists a sequence of at most + 1 proper k-*colourings*of G ... Recolouring: Changing the Height of a Vertex In order to understand the role of vertex heights in*finding**shortest**paths**between**colourings*, we investigate how the heights of vertices change along the ...##
###
Finding Shortest Paths Between Graph Colourings

2015
*
Algorithmica
*

2016) '

doi:10.1007/s00453-015-0009-7
fatcat:5wy5qyfgcbfdtl7umu3aayvdvu
*Finding**shortest**paths**between**graph**colourings*.', Algorithmica., 75 (2). pp. 295-321. Further information on publisher's website: http://dx. ... The k-*colouring*reconfiguration problem asks whether, for a given*graph*G, two proper k-*colourings*α and β of G, and a positive integer , there exists a sequence of at most + 1 proper k-*colourings*of G ... Recolouring: Changing the Height of a Vertex In order to understand the role of vertex heights in*finding**shortest**paths**between**colourings*, we investigate how the heights of vertices change along the ...##
###
An Optimal Algorithm to Find Next-to-Shortest Path between Two Vertices of Cactus Graphs

2019
*
International Journal of Research in Advent Technology
*

In this paper, we present an optimal algorithm to

doi:10.32622/ijrat.742019218
fatcat:6uxe2ruig5gsznamexrkd4fwrq
*find*a next-to-*shortest**path**between*any two vertices of a cactus*graph*with n vertices which runs in time. ... The next-to-*shortest**path**between*two vertices and is a*path*whose length is minimum among all*paths**between*and with the*shortest*ones excluded. ... the block which contains and then*find*the*shortest**path**between*and After then we*find*next-to-*shortest**path**between*two specified vertices u and v. ...##
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Finding shortest paths in the presence of orthogonal obstacles using a combined L 1 and link metric
[chapter]

1990
*
Lecture Notes in Computer Science
*

Nilsson t bengt@dna.lu.se The problem of computing

doi:10.1007/3-540-52846-6_91
fatcat:byzb246t6vfflal3nn7ifr2i2u
*shortest**paths*in obstacle environments has received considerable attention recently. ... Queries can be performed in O(log n) time, and the*shortest**path*can be reported in additional time proportional to its size. ... Most solutions to these problems first*find*a 'critical*graph*' which contains the*shortest**path**between*the source and the target. ...##
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Multicolour paths in graphs: NP-hardness, algorithms, and applications on routing in WDM networks

2016
*
Journal of combinatorial optimization
*

In this paper we study a problem of

doi:10.1007/s10878-016-0003-2
fatcat:dbkuesrlbbhb3mnhrc425bypmm
*finding**coloured**paths*of minimum weight in*graphs*. ... The problem of*finding*one or more k-multicolour*paths*in a*graph*has applications in optical network and social network analysis. ... It is also NP-hard to*find*a*shortest*(or longest)*path**between*two given vertices meeting all the*colours*in a properly*coloured*directed*graph*. ...##
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Algorithms for Finding Shortest Paths in Networks with Vertex Transfer Penalties

2020
*
Algorithms
*

In this paper we review many of the well-known algorithms for solving the

doi:10.3390/a13110269
fatcat:jmcnmx4ywrbancl2aanf3soatq
*shortest**path*problem in edge-weighted*graphs*. ... Asymptotically, compared to using Dijkstra's algorithm on expanded*graphs*, our first variant is faster for very sparse*graphs*but slower with dense*graphs*. ... These are designed to*find**shortest**paths*in our edge-*coloured**graphs*(that is,*graphs*featuring transfer penalties at their vertices), but without the need for first performing a*graph*expansion. ...##
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A Polynomial Time Algorithm for the k-Disjoint Shortest Paths Problem
[article]

2020
*
arXiv
*
pre-print

In this paper, we focus on the version of the problem where all the

arXiv:1912.10486v2
fatcat:ukxtrauxbfgk5fhit7egw3qxji
*paths*are required to be*shortest**paths*. ... The disjoint*paths*problem is a fundamental problem in algorithmic*graph*theory and combinatorial optimization. ... This means that we can*find*a*path*P ′ of*colour*i*between*two vertices x, y of S such that P ′ uses no edge of S. ...##
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On graphs coverable by k shortest paths
[article]

2022
*
arXiv
*
pre-print

We show that if the edges or vertices of an undirected

arXiv:2206.15088v1
fatcat:kzlz4fvox5f2dboilsve4k3kbu
*graph*G can be covered by k*shortest**paths*, then the pathwidth of G is upper-bounded by a function of k. ... As a corollary, we prove that the problem Isometric*Path*Cover with Terminals (which, given a*graph*G and a set of k pairs of vertices called terminals, asks whether G can be covered by k*shortest**paths*... one aims to*find*a small set of terminals of the input*graph*such that all vertices lie on a*shortest**path**between*two terminals. function on k. ...##
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Optimal Assignment of a Tree-Structured Context Reasoning Procedure onto a Host-Satellites System

2007
*
2007 IEEE International Parallel and Distributed Processing Symposium
*

problem in a doubly weighted

doi:10.1109/ipdps.2007.370327
dblp:conf/ipps/MeiPW07
fatcat:s3clym7svbgddh3o3goeqzf7cq
*graph*. ... The presented solution is a modification of an earlier method proposed by Bokhari, in which the optimal assignment problem to minimize the bottleneck processing time is transformed into a*path*-searching ... of the*shortest*-*path*(determined using S weight) is contributed by the aigoritnm Tor coiourea subsequent edges having the same*colour*, that part of the assignment*graph*should be expanded (refer to "expansion ...##
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Graphs for which the homomorphism extension reconfiguration problem is trivial
[article]

2022
*
arXiv
*
pre-print

We further give bounds on the diameter of Hom(G,H;p) in this case and show that the

arXiv:2208.04071v2
fatcat:y2ppjfquxvfabajb3x6kxtguom
*shortest**path**between*two vertices can be found in polynomial time. ... This class contains such*graphs*as chordal*graphs*and hypercubes for which the*shortest**path*reconfiguration problem has previously been shown to be trivial. ... Though we*find*the*shortest**path**between*two*shortest**paths*in an NU*graph*in polynomial time in Corollary 3.10, this does not*find*a*shortest*reconfiguraton*path*. ...##
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Open questions on Tantrix graphs

2017
*
Mathematical Gazette
*

TantrixTM tiles are black hexagons imprinted with three

doi:10.1017/mag.2017.8
fatcat:2lxqz3xyjbbu3pzjgfvbhu3ni4
*coloured**paths*[1] joining pairs of edges. There are three different kinds of*path*. ... The game is played by matching tiles to connect*paths*of the same*colour*; the goal is to create loops or long*paths*of a single*colour*This easy to learn yet hard to master game has inspired research on ... We also*find*that the girth of the*graph*, or the length of its*shortest*cycle of edges, is 4. ...##
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Finding paths between 3-colorings

2010
*
Journal of Graph Theory
*

We do so by characterising the instances G, α, β where the transformation is possible; the proof of this characterisation is via an algorithm that either

doi:10.1002/jgt.20514
fatcat:qlpx6fk7svcwblcob74n5nhahq
*finds*a sequence of recolourings*between*α and ... Given a 3-*colourable**graph*G and two proper vertex 3-*colourings*α and β of G, consider the following question : is it possible to transform α into β by recolouring vertices of G one at a time, making sure ... Then the algorithm described in Section 3*finds*a*shortest**path**between*α and β. ...##
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On the diameter of reconfiguration graphs for vertex colourings

2011
*
Electronic Notes in Discrete Mathematics
*

Lower Bounds We can show that the 3-

doi:10.1016/j.endm.2011.09.028
fatcat:agrqqtrpbjasjphw62tpvpextq
*colour*diameter of a*path*on n vertices is Θ(n 2 ) and have an example of a k-*colourable*chordal*graph*with (k + 1)-*colour*diameter of order Θ(n 2 ) for every k ≥ 4. ... Theorem 3.4 The class of chordal bipartite*graphs*is 2-*colour*-dense. ... There are exceptions such as the*shortest**path*reconfiguration problem being PSPACE-hard [1] . Also of interest is*finding**shortest**paths**between*solutions. ...##
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Properly Coloured Cycles and Paths: Results and Open Problems
[article]

2008
*
arXiv
*
pre-print

to

arXiv:0805.3901v3
fatcat:f7nre2ygwjcolaagyirqd4cp7e
*find**shortest*ones among them. ... In particular, we consider a family of transformations of an edge-*coloured*multigraph G into an ordinary*graph*that allow us to check the existence PC cycles and PC (s,t)-*paths*in G and, if they exist, ... To*find*a*shortest*PC cycle in G, choose a vertex x ∈ V (G ′ ). We will*find*a*shortest*PC cycle in G traversing x. ...
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