Hamiltonian Paths in C-shaped Grid Graphs

One of the well-known NP-complete problems in graph theory is the Hamiltonian path problem; i.e., finding a simple path in the graph such that every vertex visits exactly once [5]. The two-dimensional integer grid G is an infinite undirected graph in which vertices are all points of the plane with integer coordinates and two vertices are connected by an edge if and only if the Euclidean distance between them is equal to 1. A grid graph Gg is a finite vertex-induced subgraph of the two-dimensional integer grid G. A solid grid graph is a grid graph without holes. A rectangular grid graph R(m, n) is the subgraph of G (the infinite grid graph) induced by V(R) = {v | 1 ≤ vx ≤ m, 1 ≤ vy ≤ n}, where vx and vy are x and y coordinates of v, respectively. A C−shaped grid graph C(m, n, k, l) is a rectangular grid graph R(m, n) such that a rectangular subgraph R(k, l) is removed from it while R(m, n) and R(k, l) have exactly one border side in common, where k, l ≥ 1 and m, n > 1 (see Fig. 1(c)). In this paper, we only focus on the results on grid graphs. There are some results on Hamiltonian path for other classes of graphs which we do not mention here, see [3, 16] for more details. In [10], Itai et al. proved that the Hamiltonian path problem for general grid graphs, with or without specified endpoints, is NP-complete. They showed that the problem for rectangular grid graphs can be solved in linear time. Chen et al. [2] gave a parallel algorithm for the problem in mesh architecture. Lenhart and Umans [15] gave a polynomial-time algorithm for finding Hamiltonian cycles in solid grid graphs. Their algorithm runs in O(n4) time. Also, Salman [17] introduced a family of grid graphs, that is, alphabet grid graphs, and determined classes of alphabet grid graphs that contain Hamiltonian cycles. In [11], the authors proposed a linear-time algorithm for the Hamiltonian path problem for some small classes of grid graphs, namely L−alphabet, C−alphabet, E−alphabet, and F−alphabet grid graphs. In [14], necessary and sufficient conditions for the existence of a Hamiltonian path in L−shaped grid graphs have been studied. L−alphabet and C−alphabet grid graphs considered in [11] are special cases of L−shaped