The Hamiltonian properties of supergrid graphs

Abstract

In this paper, we first introduce a novel class of graphs, namely supergrid. Supergrid graphs include grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be NP-complete. However, they are unknown for supergrid graphs. The Hamiltonian cycle (path) problem on supergrid graphs can be applied to control the stitching traces of computerized sewing machines. In this paper, we will prove that the Hamiltonian cycle problem for supergrid graphs is NP-complete. It is easily derived from the Hamiltonian cycle result that the Hamiltonian path problem on supergrid graphs is also NP-complete. We then show that two subclasses of supergrid graphs, including rectangular (parallelism) and alphabet, always contain Hamiltonian cycles.

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

“…These propositions will be used in proving our results and are given in [13,14,15]. [13,14,15]) Let C 1 and C 2 be two vertex-disjoint cycles of a graph G, let C 1 and P 1 be a cycle and a path, respectively, of G with V (C 1 ) ∩ V (P 1 ) = ∅, and let x be a vertex in G − V (C 1 ) or G − V (P 1 ). Then, the following statements hold true: (1) If there exist two edges e 1 ∈ C 1 and e 2 ∈ C 2 such that e 1 ≈ e 2 , then C 1 and C 2 can be combined into a cycle of G (see Fig.…”

“…It is well known that the Hamiltonian and longest (s, t)-path problems are NP-complete for general graphs [7,22]. The same holds true for bipartite graphs [32], split graphs [8], circle graphs [6], undirected path graphs [1], grid graphs [21], triangular grid graphs [9], supergrid graphs [13], and so on. In the literature, there are many studies for the Hamiltonian connectivity of interconnection networks, see [3,5,10,11,12,34,35,36].…”

“…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. Since the Hamiltonian cycle and path problems are NP-complete for supergrid graphs [13], an important line of investigation is to discover the complexities of the Hamiltonian related problems when the input is restricted to be in special subclasses of supergrid graphs.…”

“…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. Since the Hamiltonian cycle and path problems are NP-complete for supergrid graphs [13], an important line of investigation is to discover the complexities of the Hamiltonian related problems when the input is restricted to be in special subclasses of supergrid graphs. In [14], we showed that the Hamiltonian cycle problem for linear-convex supergrid graphs is linear solvable.…”

Hamiltonian cycles in linear-convex supergrid graphs

“…It is well known that the Hamiltonian problems are NP-complete for general graphs [10,21]. The same holds true for bipartite graphs [24], split graphs [11], circle graphs [7], undirected path graphs [3], planar bipartite graphs with maximum degree 3 [20], grid graphs [20], triangular grid graphs [12], and supergrid graphs [18].…”