Reconfiguring Simple s, t Hamiltonian Paths in Rectangular Grid Graphs

2021 ◽  
pp. 501-515
Author(s):  
Rahnuma Islam Nishat ◽  
Venkatesh Srinivasan ◽  
Sue Whitesides
Keyword(s):  
Rectangular Grid ◽  
Hamiltonian Paths ◽  
Grid Graphs

scholarly journals A linear-time algorithm for the longest path problem in rectangular grid graphs

2012 ◽  
Vol 160 (3) ◽  
pp. 210-217 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri ◽  
Asghar Asgharian-Sardroud
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Rectangular Grid ◽  
Grid Graphs ◽  
Longest Path ◽  
Longest Path Problem

scholarly journals Hamiltonian Paths in Some Classes of Grid Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Grid Graphs ◽  
Hamiltonian Path Problem ◽  
Np Complete ◽  
Necessary And Sufficient ◽  
Linear Time Algorithms

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.

scholarly journals NP-completeness and One Polynomial Subclass of the Two-Step Graph Colouring Problem

2019 ◽  
Vol 26 (3) ◽  
pp. 405-419
Author(s):  
Natalya Sergeevna Medvedeva ◽  
Alexander Valeryevich Smirnov
Keyword(s):  
Connected Graph ◽  
Special Interest ◽  
Recognition Problem ◽  
Graph Colouring ◽  
Rectangular Grid ◽  
Grid Graph ◽  
Grid Graphs ◽  
Np Completeness ◽  
Polynomial Reduction ◽  
Vertex Colouring

In this paper, we study the two-step colouring problem for an undirected connected graph. It is required to colour the graph in a given number of colours in a way, when no pair of vertices has the same colour, if these vertices are at a distance of 1 or 2 between each other. Also the corresponding recognition problem is set. The problem is closely related to the classical graph colouring problem. In the article, we study and prove the polynomial reduction of the problems to each other. So it allows us to prove NP-completeness of the problem of two-step colouring. Also we specify some of its properties. Special interest is paid to the problem of two-step colouring in application to rectangular grid graphs. The maximum vertex degree in such a graph is between 0 and 4. For each case, we elaborate and prove the function of two-vertex colouring in the minimum possible number of colours. The functions allow each vertex to be coloured independently from others. If vertices are examined in a sequence, colouring time is polynomial for a rectangular grid graph.

scholarly journals Enumerating Hamiltonian Cycles

10.37236/4510 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Ville H. Pettersson
Keyword(s):  
Dynamic Programming ◽  
Three Dimensional ◽  
Programming Method ◽  
Hamiltonian Cycles ◽  
Dynamic Programming Method ◽  
Hamiltonian Paths ◽  
Grid Graphs ◽  
Triangular Grids ◽  
Arbitrary Graphs

A dynamic programming method for enumerating hamiltonian cycles in arbitrary graphs is presented. The method is applied to grid graphs, king's graphs, triangular grids, and three-dimensional grid graphs, and results are obtained for larger cases than previously published. The approach can easily be modified to enumerate hamiltonian paths and other similar structures.

scholarly journals The number of Hamiltonian paths in a rectangular grid

1997 ◽  
Vol 169 (1-3) ◽  
pp. 29-38 ◽  
Author(s):  
Karen L. Collins ◽  
Lucia B. Krompart
Keyword(s):  
Rectangular Grid ◽  
Hamiltonian Paths

scholarly journals Hamiltonian paths in L-shaped grid graphs

2016 ◽  
Vol 621 ◽  
pp. 37-56 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri
Keyword(s):  
Hamiltonian Paths ◽  
Grid Graphs

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

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