The Hamiltonian connectivity of rectangular supergrid graphs

Hung, Ruo-Wei; Li, Chin-Feng; Chen, Jong-Shin; Su, Qingsong

doi:10.1016/j.disopt.2017.06.001

Cited by 16 publications

(68 citation statements)

References 38 publications

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“…In addition to condition (F1) (as depicted in Fig. 5(a) and 5(b)), in [17,31], we showed that HP (L(m, n; k, l), s, t) does not exist whenever one of the following conditions is satisfied.…”

Section: Terminologies and Background Resultsmentioning

confidence: 78%

“…In [15], the authors showed that HP (R(m, n), s, t) does not exist if the following condition holds:…”

Section: Terminologies and Background Resultsmentioning

confidence: 99%

“…6(c)), in [19], we showed that C(m, n; k, l; c, d) contains no Hamiltonian (s, t)-path if (C(m, n; k, l; c, d), s, t) satisfies one of the following conditions. The Hamiltonian (s, t)-path P of R(m, n) constructed in [15] satisfies that P contains at least one boundary edge of each boundary, and is called canonical. [15]) Let R(m, n) be a rectangular supergrid graph with m, n 1, and let s and t be its two distinct vertices.…”

Section: Terminologies and Background Resultsmentioning

confidence: 99%

“…Recently, we showed the Hamiltonian connectivity of alphabet supergrid graphs [18]. Very recently, we showed that the Hamiltonian and longest (s, t)-paths of L-and C-shaped supergrid graphs can be computed in linear time [17,31,19]. In this paper, we first show that O-shaped supergrid graphs, which are rectangular supergrid graphs with rectangular holes, are always Hamiltonian and Hamiltonian connected except few conditions.…”

mentioning

confidence: 81%

“…In [14], we showed that the Hamiltonian cycle problem for linear-convex supergrid graphs is linear solvable. In [15], we proved that rectangular supergrid graphs are always Hamiltonian connected except one trivial forbidden condition. Some shaped supergrid graphs have been verified to be Hamiltonian and Hamiltonian connected [16].…”

mentioning

confidence: 99%

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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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Section: Terminologies and Background Resultsmentioning

confidence: 78%

“…In [15], the authors showed that HP (R(m, n), s, t) does not exist if the following condition holds:…”

Section: Terminologies and Background Resultsmentioning

confidence: 99%