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### Finding Shortest Paths between Graph Colourings [article]

Matthew Johnson and Dieter Kratsch and Stefan Kratsch and Viresh Patel and Daniël Paulusma
2014 arXiv   pre-print
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 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  ...

### Finding Shortest Paths Between Graph Colourings [chapter]

Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, Daniël Paulusma
2014 Lecture Notes in Computer Science
2016) '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

Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, Daniël Paulusma
2015 Algorithmica
2016) '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

Sambhu Charan Barman,, Madhumangal Pal, Sukumar Mondal
2019 International Journal of Research in Advent Technology
In this paper, we present an optimal algorithm to 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.  ...

### Finding shortest paths in the presence of orthogonal obstacles using a combined L 1 and link metric [chapter]

Mark Berg, Marc Kreveld, Bengt J. Nilsson, Mark H. Overmars
1990 Lecture Notes in Computer Science
Nilsson t bengt@dna.lu.se The problem of computing 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.  ...

### Multicolour paths in graphs: NP-hardness, algorithms, and applications on routing in WDM networks

Rafael F. Santos, Alessandro Andrioni, Andre C. Drummond, Eduardo C. Xavier
2016 Journal of combinatorial optimization
In this paper we study a problem of 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.  ...

### Algorithms for Finding Shortest Paths in Networks with Vertex Transfer Penalties

Rhyd Lewis
2020 Algorithms
In this paper we review many of the well-known algorithms for solving the 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.  ...

### A Polynomial Time Algorithm for the k-Disjoint Shortest Paths Problem [article]

William Lochet
2020 arXiv   pre-print
In this paper, we focus on the version of the problem where all the 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.  ...

### On graphs coverable by k shortest paths [article]

Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca
2022 arXiv   pre-print
We show that if the edges or vertices of an undirected 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.  ...

### Optimal Assignment of a Tree-Structured Context Reasoning Procedure onto a Host-Satellites System

Hailiang Mei, Pravin Pawar, Ing Widya
2007 2007 IEEE International Parallel and Distributed Processing Symposium
problem in a doubly weighted 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  ...

### Graphs for which the homomorphism extension reconfiguration problem is trivial [article]

Mark Siggers
2022 arXiv   pre-print
We further give bounds on the diameter of Hom(G,H;p) in this case and show that the 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.  ...

### Open questions on Tantrix graphs

Heidi Burgiel, Mahmoud El-Hashash
2017 Mathematical Gazette
TantrixTM tiles are black hexagons imprinted with three coloured paths  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.  ...

### Finding paths between 3-colorings

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

### On the diameter of reconfiguration graphs for vertex colourings

Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, Daniël Paulusma
2011 Electronic Notes in Discrete Mathematics
Lower Bounds We can show that the 3-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  . Also of interest is finding shortest paths between solutions.  ...

### Properly Coloured Cycles and Paths: Results and Open Problems [article]

Gregory Gutin, Eun Jung Kim
2008 arXiv   pre-print
to 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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