The Hamiltonian Connected Property of Some Shaped Supergrid Graphs

A Hamiltonian path (cycle) of a graph is a simple path (cycle) which visits each vertex of the graph exactly once. The Hamiltonian path (cycle) problem is to determine whether a graph contains a Hamiltonian path (cycle). A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. Supergrid graphs were first introduced by us and include grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian path (cycle) problem for grid graphs and triangular grid graphs was known to be NP-complete. Recently, we have proved that they are also NP-complete for supergrid graphs. These problems on supergrid graphs can be applied to control the stitching traces of computerized sewing machines. Very recently, we showed that rectangular supergrid graphs are Hamiltonian connected except two trivial forbidden conditions. In this paper, we will study the Hamiltonian connectivity of some shaped supergrid graphs, including triangular, parallelogram, and trapezoid. We prove that these shaped supergrid graphs are always Hamiltonian connected except few trivial forbidden conditions.