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

2022
*
arXiv
*
pre-print

A

arXiv:2205.08025v1
fatcat:akseq5h7tjbb5hgymx2gjxeuam
*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*...##
###
The Hamiltonian connectivity of rectangular supergrid graphs

2017
*
Discrete Optimization
*

A

doi:10.1016/j.disopt.2017.06.001
fatcat:c7um4x3euzggpg4nzibjwvf3pq
*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. ...##
###
Hamiltonian Paths in Some Classes of Grid Graphs

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. ...

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

2019
*
arXiv
*
pre-print

Supergrid

arXiv:1904.02581v2
fatcat:wa34o3dfwbcqfptetu6wa57z3m
*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. ...##
###
An efficient parallel algorithm for the longest path problem in meshes
[article]

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] . ...

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

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. ...

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

2021
*
arXiv
*
pre-print

As

arXiv:2110.08325v1
fatcat:xqa7a2dtvvgopgsrrq4mna4pxe
*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. ...##
###
The Hamiltonian properties of supergrid graphs

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. ...

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

2014
*
arXiv
*
pre-print

Our method

arXiv:1410.2388v1
fatcat:rtmgh4er5zee5bqngilz45jy4a
*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. ...##
###
Spanning closed trail and hamiltonian cycle in grid graphs
[chapter]

1995
*
Lecture Notes in Computer Science
*

Finally we show that a

doi:10.1007/bfb0015440
fatcat:x7sscv5pt5bozoxfwcqyg5yfyq
*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. ...##
###
The Complexity of Snake

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

*s*∈

*S*) that use more than two blue

*paths*

*in*(G \

*S*P ) ∪

*s*. ...

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

2022
*
Algorithms
*

A

doi:10.3390/a15020061
fatcat:pyn77bivkferhhjdnrbwmsirgu
*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. ...##
###
Finding Hamiltonian and Longest (s, t)-paths of C-shaped Supergrid Graphs in Linear Time
[article]

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. ...

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

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. ...

##
###
Hamiltonian Paths in C-shaped Grid Graphs
[article]

2016
*
arXiv
*
pre-print

We study

arXiv:1602.07407v1
fatcat:mvadcr3narapfnmvj7udpc25wi
*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*. ...
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