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

Hung, Ruo-Wei; Keshavarz-Kohjerdi, Fatemeh

doi:10.48550/arxiv.1908.07447

Cited by 1 publication

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“…6(a)-6(b)) and (F2) (as depicted in Fig. 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. (g)…”

Section: Terminologies and Background Resultsmentioning

confidence: 76%

“…In [17] and [19], we verified the Hamiltonicity of L-shaped and C-shaped supergrid graphs as follows.…”

Section: Terminologies and Background Resultsmentioning

confidence: 99%

“…Theorem 2.2. (See [15,17,31,19]) Let G be a supergrid graph with vertices s and t, where G is rectangular, Lshaped, or C-shaped. (G, s, t) contains a Hamiltonian (s, t)-path, i.e., HP (G, s, t) does exist if and only if (G, s, t) does not satisfy conditions (F1)-(F6).…”

Section: Terminologies and Background Resultsmentioning

confidence: 99%

“…Lemma 2.5. (See [19]) Let R(m, 3) be a 3-rectangle with m 3, and let s and t be its two distinct vertices. Let z 1 = (m, 1), z 2 = (m, 2), and z 3 = (m, 3) be three vertices of R(m, 3), and let edges e 12 = (z 1 , z 2 ), e 23 = (z 2 , z 3 ).…”

Section: Terminologies and Background Resultsmentioning

confidence: 99%

“…For the longest (s, t)-path problem on R(m, n), L(m, n; k, l), and C(m, n; k, l; c, d), we showed in [15,31,19] that it can be solved in linear time. In this paper, we will study O-shaped supergrid graph O(m, n; k, l; a, b, c, d) whose structure is depicted in Fig.…”

Section: Terminologies and Background Resultsmentioning

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: 76%

“…In [17] and [19], we verified the Hamiltonicity of L-shaped and C-shaped supergrid graphs as follows.…”

Section: Terminologies and Background Resultsmentioning

confidence: 99%