A linear-time algorithm for finding Hamiltonian (s, t)-paths in odd-sized rectangular grid graphs with a rectangular hole

Keshavarz-Kohjerdi, Fatemeh; Bagheri, Alireza

doi:10.1007/s11227-017-1984-z

Cited by 12 publications

(2 citation statements)

References 30 publications

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“…They also presented a linear-time algorithm for computing the longest path between two given vertices in rectangular grid graphs [22], gave a parallel algorithm to solve the longest path problem in rectangular grid graphs [23], and solved the Hamiltonian connected problem in L-shaped grid graphs [24]. Very recently, Keshavarz-Kohjerdi and Bagheri presented a linear-time algorithm to find Hamiltonian (s, t)-paths in rectangular grid graphs with a rectangular hole [25,26]. Reay and Zamfirescu [32] proved that all 2-connected, linear-convex triangular grid graphs except one special case contain Hamiltonian cycles.…”

Section: Introductionmentioning

confidence: 99%

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

Keshavarz-Kohjerdi¹,

Hung²

2019

Preprint

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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. In this paper, we first prove that every L-shaped supergrid graph always contains a Hamiltonian cycle except one trivial condition. We then verify the Hamiltonian connectivity of L-shaped supergrid graphs except few conditions. The Hamiltonicity and Hamiltonian connectivity of L-shaped supergrid graphs can be applied to compute the minimum trace of computerized embroidery machine and 3D printer when a L-like object is printed. Finally, we present a linear-time algorithm to compute the longest (s, t)-path of L-shaped supergrid graph given two distinct vertices s and t.

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Section: Introductionmentioning

confidence: 99%

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

Keshavarz-Kohjerdi¹,

Hung²

2019

Preprint

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“…Recently, verified the Hamiltonian connectivity of L-shaped grid graphs. Very recently, Keshavarz-Kohjerdi and Bagheri presented a linear-time algorithm to find Hamiltonian (s, t)-paths in rectangular grid graphs with a rectangular hole [27,28], and to compute longest (s, t)-paths in L-shaped and C-shaped grid graphs [29,30]. The supergrid graphs were first appeared in [13], in which we proved that the Hamiltonian cycle and path problems on supergrid graphs are NP-complete, and every rectangular supergrid graph is Hamiltonian.…”

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confidence: 99%

Finding Longest (s, t)-paths of O-shaped Supergrid Graphs in Linear Time

Hung

Keshavarz-Kohjerdi

Tseng

et al. 2019

2019 IEEE 10th International Conference on Awareness Science and Technology (iCAST)

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In this paper, we continue the study of the Hamiltonian and longest (s, t)-paths of supergrid graphs. 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 to t in the graph. 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. An O-shaped supergrid graph is a special kind of a rectangular grid graph with a rectangular hole. In this paper, we first prove the Hamiltonian connectivity of O-shaped supergrid graphs except few conditions. We then show that the longest (s, t)-path of an O-shaped supergrid graph can be computed in linear time. The Hamiltonian and longest (s, t)-paths of O-shaped supergrid graphs can be applied to compute the minimum trace of computerized embroidery machine and 3D printer when a hollow object is printed.Hamiltonian path and cycle problems for grid graphs are NP-complete. They also gave the necessary and sufficient conditions for a rectangular grid graph to be Hamiltonian connected. Thus, rectangular grid graphs are not always Hamiltonian connected. Zamfirescu et al. [39] gave the sufficient conditions for a grid graph having a Hamiltonian cycle, and proved that all grid graphs of positive width have Hamiltonian line graphs. Later, Chen et al. [4] improved the Hamiltonian path algorithm of [21] on rectangular grid graphs and presented a parallel algorithm for the Hamiltonian path problem with two given end vertices in rectangular grid graph. Also Lenhart and Umans [33] showed the Hamiltonian cycle problem on solid grid graphs, which are grid graphs without holes, is solvable in polynomial time. Recently, 25] presented linear-time and parallel algorithms to compute the longest path between two given vertices in rectangular grid graphs. Reay and Zamfirescu [37] proved that all 2-connected, linearconvex triangular grid graphs contain Hamiltonian cycles except one special case. The Hamiltonian cycle and path problems on triangular grid graphs were known to be NP-complete [9]. In addition, the Hamiltonian cycle problem on hexagonal grid graphs has been shown to be NP-complete [20]. Alphabet grid graphs first appeared in [38], in which Salman determined the classes of alphabet grid graphs containing Hamiltonian cycles. Keshavarz-Kohjerdi and Bagheri [23] gave the necessary and sufficient conditions for the existence of Hamiltonian paths in alphabet grid graphs, and presented a linear-time algorithm for finding Hamiltonian path with two given endpoints in these graphs. Recently, verified the Hamiltonian connectivity of L-shaped grid graphs. Very recently, Keshavarz-Kohjerdi and Bagheri presented a linear-time algorithm to find Hamiltonian (s, t)-paths in rectangular grid graphs with a rectangular hole [27,28], and to compute longest (s, t)-paths in L-shaped and C-shaped grid graphs [29,30]. The supergrid graphs were first appeared in [13]...

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Embedding linear arrays of the maximum length in O-shaped meshes

Keshavarz-Kohjerdi

2021

J Supercomput

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A linear-time algorithm for finding Hamiltonian (s, t)-paths in odd-sized rectangular grid graphs with a rectangular hole

Cited by 12 publications

References 30 publications

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

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

Finding Longest (s, t)-paths of O-shaped Supergrid Graphs in Linear Time

Embedding linear arrays of the maximum length in O-shaped meshes

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