The Hamiltonian Connectivity of Alphabet Supergrid Graphs ⋆ Ruo-Wei Hung 1,* , Fatemeh Keshavarz-Kohjerdi 2 , Chuan-Bi Lin 3 , and Jong-Shin Chen 3 Abstract—The Hamiltonian path problem on general graphs is well-known to be NP-complete. In the past, we have proved it to be also NP-complete for supergrid graphs. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices in it. Determining whether a supergrid graph is Hamiltonian connected is clear to be NP- complete. Recently, we proved the Hamiltonian connectivity of some special supergrid graphs, including rectangular, triangu- lar, parallelogram, and trapezoid. In this paper, we will study the Hamiltonian connectivity of alphabet supergrid graphs. There are 26 types of alphabet supergrid graphs in which every capital letter is represented by a type of alphabet supergrid graphs. We will prove L-, C-, F-, E-, N-, and Y-alphabet supergrid graphs to be Hamiltonian connected. The Hamil- tonian connectivity of the other alphabet supergrid graphs can be verified similarly. The Hamiltonian connected property of alphabet supergrid graphs can be applied to compute the minimum stitching trace of computerized embroidery machines during the sewing process. Index Terms—Hamiltonian connectivity, alphabet supergrid graphs, shaped supergrid graphs, computerized embroidery machines. I. I NTRODUCTION A Hamiltonian path (resp., cycle) of a graph is a simple path (resp., cycle) in which each vertex of the graph appears exactly once. The Hamiltonian path (resp., cycle) problem is to determine whether or not a graph contains a Hamiltonian path (resp., cycle). A graph G is said to be Hamiltonian if it contains a Hamiltonian cycle, and is called Hamiltonian connected if for each pair of distinct vertices u and v of G, there exists a Hamiltonian path between u and v in G. The Hamiltonian path and cycle problems have numerous applications in different areas, including establishing transport routes, production launching, the on- line optimization of flexible manufacturing systems [1], com- puting the perceptual boundaries of dot patterns [40], pattern recognition [2], [42], [45], DNA physical mapping [14], fault-tolerant routing for 3D network-on-chip architectures [9], etc. It is well known that the Hamiltonian path and cycle problems are NP-complete for general graphs [11], Manuscript received August 07, 2018; revised December 25, 2018. This work was supported in part by the Ministry of Science and Technology, Taiwan under grant no. MOST 105-2221-E-324-010-MY3. ⋆ A preliminary version of this paper has appeared in: 2017 IEEE 8th International Conference on Awareness Science and Technology (iCAST 2017), Taichung, Taiwan, 2017, pp. 27–34 [21]. 1 Ruo-Wei Hung is with the Department of Computer Science and Information Engineering, Chaoyang University of Technology, Wufeng, Taichung 41349, Taiwan. 2 Fatemeh Keshavarz-Kohjerdi is with the Department of Computer Sci- ence, Shahed University, Tehran, Iran. 3 Chuan-Bi Lin and Jong-Shin Chen are with the Department of Informa- tion and Communication Engineering, Chaoyang University of Technology, Wufeng, Taichung 41349, Taiwan. * Corresponding author e-mail: rwhung@cyut.edu.tw. [28]. The same holds true for bipartite graphs [35], split graphs [12], circle graphs [8], undirected path graphs [3], grid graphs [27], triangular grid graphs [13], and supergrid graphs [17]. In the literature, there are many studies for the Hamiltonian connectivity of interconnection networks, including WK-recursive network [10], recursive dual-net [37], hypercomplete network [5], alternating group graph [29], arrangement graph [39], augmented hypercube [16], generalized base-b hypercube [23], hyercube-like network [41], twisted cube [25], crossed cube [24], M¨ obius cube [7], folded hypercube [15], and enhanced hypercube [38]. In this paper, we will verify the Hamiltonian connectivity of alphabet supergrid graphs. The two-dimensional integer grid graph G ∞ is an infinite graph whose vertex set consists of all points of the Euclidean plane with integer coordinates and in which two vertices are adjacent if the (Euclidean) distance between them is equal to 1. The two-dimensional triangular grid graph T ∞ is an infinite graph obtained from G ∞ by adding all edges on the lines traced from up-left to down-right. A grid graph is a finite, vertex-induced subgraph of G ∞ . For a node v in the plane with integer coordinates, let v x and v y represent its x and y coordinates, respectively, denoted by v =(v x ,v y ). If v is a vertex in a grid graph, then its possible adjacent vertices include (v x ,v y −1), (v x −1,v y ), (v x +1,v y ), and (v x ,v y +1). A triangular grid graph is a finite, vertex-induced sub- graph of T ∞ . If v is a vertex in a triangular grid graph, then its possible neighboring vertices include (v x ,v y − 1), (v x − 1,v y ), (v x +1,v y ), (v x ,v y + 1), (v x − 1,v y − 1), and (v x +1,v y + 1). Thus, triangular grid graphs contain grid graphs as subgraphs. For example, Fig. 1(a) and Fig. 1(b) depict a grid graph and a triangular graph, respectively. The triangular grid graphs defined above are isomorphic to the original triangular grid graphs in [13] but these graphs are different when considered as geometric graphs. By the same construction of triangular grid graphs obtained from grid graphs, we defined a new class of graphs, namely supergrid graphs, in [17]. The two-dimensional supergrid graph S ∞ is an infinite graph obtained from T ∞ by adding all edges on the lines traced from up-right to down-left. That is, S ∞ is the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates is not larger than 1. A supergrid graph is a finite, vertex-induced subgraph of S ∞ . We will color vertex v white if v x + v y ≡ 0 (mod 2); otherwise, v is colored black. The possible adjacent vertices of a vertex v =(v x ,v y ) in a supergrid graph hence include (v x ,v y − 1), (v x − 1,v y ), (v x +1,v y ), (v x ,v y + 1), (v x − 1,v y − 1), (v x +1,v y + 1), (v x +1,v y − 1), and (v x − 1,v y + 1). Then, supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. For instance, Fig. IAENG International Journal of Applied Mathematics, 49:1, IJAM_49_1_10 (Advance online publication: 1 February 2019) ______________________________________________________________________________________