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The Hamiltonian Path Graph is Connected for Simple s,t Paths in Rectangular Grid Graphs [article]

Rahnuma Islam Nishat, Venkatesh Srinivasan, Sue Whitesides
2022 arXiv   pre-print
A simple s,t path P in a rectangular grid graph 𝔾 is a Hamiltonian path from the top-left corner s to the bottom-right corner t such that each internal subpath of P with both endpoints a and b on the  ...  Our reconfiguration result proves that the Hamiltonian path graph G for simple s,t paths is connected and has diameter at most 5|𝔾|/4 which is asymptotically tight.  ...  They used two operations flip and transpose, and showed that the Hamiltonian cycle graph with respect to those two operations is connected for 1-complex cycles in rectangular grids and L-shaped grid graphs  ... 
arXiv:2205.08025v1 fatcat:akseq5h7tjbb5hgymx2gjxeuam

The Hamiltonian connectivity of rectangular supergrid graphs

Ruo-Wei Hung, Chin-Feng Li, Jong-Shin Chen, Qing-Song Su
2017 Discrete Optimization  
A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path.  ...  In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs.  ...  INTRODUCTION A Hamiltonian path in a graph is a simple path in which each vertex of the graph appears exactly once. A Hamiltonian cycle in a graph is a simple cycle with the same property.  ... 
doi:10.1016/j.disopt.2017.06.001 fatcat:c7um4x3euzggpg4nzibjwvf3pq

Hamiltonian Paths in Some Classes of Grid Graphs

Fatemeh Keshavarz-Kohjerdi, Alireza Bagheri
2012 Journal of Applied Mathematics  
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.  ...  Introduction Hamiltonian path in a graph is a simple path that visits every vertex exactly once.  ... 
doi:10.1155/2012/475087 fatcat:n36tsn7awjeapkeox3dw353vvi

The Hamiltonicity, Hamiltonian Connectivity, and Longest (s, t)-path of L-shaped Supergrid Graphs [article]

Fatemeh Keshavarz-Kohjerdi, Ruo-Wei Hung
2019 arXiv   pre-print
Supergrid graphs contain grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for general supergrid graphs were known to be NP-complete.  ...  A graph is called Hamiltonian if it contains a Hamiltonian cycle, and is said to be Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices in it.  ...  Acknowledgments This work is partly supported by the Ministry of Science and Technology, Taiwan under grant no. MOST 105-2221-E-324-010-MY3.  ... 
arXiv:1904.02581v2 fatcat:wa34o3dfwbcqfptetu6wa57z3m

An efficient parallel algorithm for the longest path problem in meshes [article]

Fatemeh Keshavarz-Kohjerdi, Alireza Bagheri
2012 arXiv   pre-print
In this paper, first we give a sequential linear-time algorithm for the longest path problem in meshes. This algorithm can be considered as an improvement of [13].  ...  Then based on this sequential algorithm, we present a constant-time parallel algorithm for the problem which can be run on every parallel machine.  ...  For constructing a Hamiltonian path in a rectangular grid graph, we use the constant-time algorithm of [3] .  ... 
arXiv:1201.4459v1 fatcat:6e4h22a56vfsxksey6yhac5od4

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

Fatemeh Keshavarz-Kohjerdi, Alireza Bagheri, Asghar Asgharian-Sardroud
2012 Discrete Applied Mathematics  
In this paper, we give a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph.  ...  The longest path problem is a well-known NP-hard problem and so far it has been solved polynomially only for a few classes of graphs.  ...  Acknowledgments The authors would like to thank the referees for a very careful reading and many helpful comments and suggestions, which have led to great improvement of the paper.  ... 
doi:10.1016/j.dam.2011.08.010 fatcat:c7zfwxnmqrfmbbizt6tkueucnu

Minor Embedding in Broken Chimera and Pegasus Graphs is NP-complete [article]

Elisabeth Lobe, Annette Lutz
2021 arXiv   pre-print
As the Hamiltonian cycle problem is a special case of the embedding problem, this proves the general complexity result for the Chimera graphs.  ...  In this work we show the hardness of the embedding problem for both types of existing hardware, represented by the Chimera and the Pegasus graphs, containing unavailable qubits.  ...  Let T e be the grid tentacle between S v and S w representing edge e = vw ∈ E(B) in L(B). W.l.o.g. for v being even and w being odd, T e is connected to S v by two edges and to S w by a single edge.  ... 
arXiv:2110.08325v1 fatcat:xqa7a2dtvvgopgsrrq4mna4pxe

The Hamiltonian properties of supergrid graphs

Ruo-Wei Hung, Chih-Chia Yao, Shang-Ju Chan
2015 Theoretical Computer Science  
In this paper, we will prove that the Hamiltonian cycle problem for supergrid graphs is NP-complete.  ...  The Hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be NP-complete. However, they are unknown for supergrid graphs.  ...  This work is partly supported by the Ministry of Science and Technology of R.O.C. (Taiwan) under grant no. MOST 104-2221-E-324-003.  ... 
doi:10.1016/j.tcs.2015.08.024 fatcat:lwfu33k6fjf63avwavvd22sb2u

Unit-length embedding of cycles and paths on grid graphs [article]

Asghar Asgharian Sardroud, Alireza Bagheri
2014 arXiv   pre-print
Our method for finding cycle of length k in rectangular grid graphs also introduces a linear-time algorithm for finding cycles of a given length k in hamiltonian solid grid graphs.  ...  We give the necessary and sufficient conditions for the existence of cycles of given length k and paths of given length k between two given vertices in n-vertex rectangular grid graphs and introduce two  ...  Lemma 4. 1 . 1 In a rectangular grid graph, a path between any two vertices s and t is shortest if and only if it is monotone.  ... 
arXiv:1410.2388v1 fatcat:rtmgh4er5zee5bqngilz45jy4a

Spanning closed trail and hamiltonian cycle in grid graphs [chapter]

Hwan-Gue Cho, Alexander Zelikovsky
1995 Lecture Notes in Computer Science  
Finally we show that a graph product G to a simple path with length n, G Pn, is hamiltonian for all n 2, if G is a polymino with a perfect matching. ?  ...  For general grid graphs we prove that the problem of nding that trail is NP-complete and for a wide subclass of grid graphs, called polymino, we give an optimal algorithm if it exists.  ...  They gave a necessary and su cient condition for the existence of the hamiltonian path in rectangular grids 6]. Also they have shown that the hamiltonian problem for general grids is NP-complete.  ... 
doi:10.1007/bfb0015440 fatcat:x7sscv5pt5bozoxfwcqyg5yfyq

The Complexity of Snake

Marzio De Biasi, Tim Ophelders, Marc Herbstritt
2016 Fun with Algorithms  
In particular, we prove the NP-hardness of collecting all food on solid grid graphs; as well as its PSPACE-completeness on general grid graphs.  ...  We extend this framework for the analysis of puzzles whose initial state is chosen by the player.  ...  Observe that there are no simple paths (connecting the two connector ports at the ends of sS) that use more than two blue paths in (G \ S P ) ∪ s.  ... 
doi:10.4230/lipics.fun.2016.11 dblp:conf/fun/BiasiO16 fatcat:zji33yjxtrdilj3jumg2liin44

Finding Hamiltonian and Longest (s,t)-Paths of C-Shaped Supergrid Graphs in Linear Time

Fatemeh Keshavarz-Kohjerdi, Ruo-Wei Hung
2022 Algorithms  
A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices.  ...  In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs.  ...  Acknowledgments: The authors would like to thank anonymous referees for many useful comments and suggestions which have improved the presentation and correctness of this paper.  ... 
doi:10.3390/a15020061 fatcat:pyn77bivkferhhjdnrbwmsirgu

Finding Hamiltonian and Longest (s, t)-paths of C-shaped Supergrid Graphs in Linear Time [article]

Ruo-Wei Hung, Fatemeh Keshavarz-Kohjerdi
2019 arXiv   pre-print
The Hamiltonian path (cycle) problem is to determine whether a graph contains a simple path (cycle) in which each vertex of the graph appears exactly once.  ...  A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices.  ...  Acknowledgments This work is partly supported by the Ministry of Science and Technology, Taiwan under grant no. MOST 105-2221-E-324-010-MY3.  ... 
arXiv:1908.07447v1 fatcat:agdikrcq2fgy7juuokiozwtpka

The Longest (s, t)-paths of O-shaped Supergrid Graphs [article]

Ruo-Wei Hung, Fatemeh Keshavarz-Kohjerdi
2019 arXiv   pre-print
The Hamiltonian (s, t)-path of a graph is a Hamiltonian path between any two given vertices s and t in the graph, and the longest (s, t)-path is a simple path with the maximum number of vertices from s  ...  A graph holds Hamiltonian connected property if it contains a Hamiltonian (s, t)-path. These two problems are well-known NP-complete for general supergrid graphs.  ...  Acknowledgments This work is partly supported by the Ministry of Science and Technology, Taiwan under grant no. MOST 108-2221-E-324-012-MY2.  ... 
arXiv:1911.08558v1 fatcat:f5gh3rqrfzdzhbtxypgvtfiqa4

Hamiltonian Paths in C-shaped Grid Graphs [article]

Fatemeh Keshavarz-Kohjerdi, Alireza Bagheri
2016 arXiv   pre-print
We study the Hamiltonian path problem in C-shaped grid graphs, and present the necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in these graphs.  ...  We also give a linear-time algorithm for finding a Hamiltonian path between two given vertices of a C-shaped grid graph, if it exists.  ...  The C−shaped grid graphs in which there is no Hamiltonian (s, t)−path. Figure 8 . 8 The C−shaped grid graphs in which there is no Hamiltonian (s, t)−path.  ... 
arXiv:1602.07407v1 fatcat:mvadcr3narapfnmvj7udpc25wi
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