Hamiltonian properties of triangular grid graphs (original) (raw)

Cyclic properties of triangular grid graphs

Proceedings of the 12th IFAC Symposium on Information Control Problems in Manufacturing, 17 - 19 May 2006, Vol. 3, 149 - 153., 2006

It is known that all 2-connected, linearly convex triangular grid graphs, with only one exception, are hamiltonian . In the paper, it is shown that this result holds for a wider class of connected, locally connected triangular grid graphs and, with more exceptions, even for some general class of graphs. It is also shown that the HAMILTONIAN CYCLE problem is NP-complete for triangular grid graphs.

Hamiltonicity of Topological Grid Graphs

J. Univers. Comput. Sci., 2007

In this paper we study connectivity and hamiltonicity properties of the topological grid graphs, which are a natural type of planar graphs associated with finite subgraphs of the usual square lattice graph of the plane. The main results are as follows. The shortness coefficient of the family of all topological grid graphs is at most 16/17. Every 3-connected topological grid graph is hamiltonian.

Some new characterizations of Hamiltonian cycles in triangular grid graphs

In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate the thermodynamics of protein folding. In this paper, we present new characterizations of the Hamiltonian cycles in labeled triangular grid graphs, which are graphs constructed from rectangular grids by adding a diagonal to each cell. By using these characterizations and implementing the computational method outlined here, we confirm the existing data, and obtain some new results that have not been published. A new interpretation of Catalan numbers is also included.

COMPLEXITY OF THE HAMILTONIAN CYCLE PROBLEM IN TRIANGULAR GRID GRAPHS

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. We show that the problem Hamil- tonian Cycle is NP-complete for triangular grid graphs, while a hamiltonian cycle in connected, locally connected triangular grid graph can be found in polynomial time. 2000 Mathematics Subject Classification: 05C38 (05C45, 68Q25). In this paper, we prove that the Hamiltonian Cycle problem is NP-complete for tri- angular grid graphs, while a hamiltonian cycle in connected, locally connected triangular grid graph can be found in polynomial time. A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. Some hamiltonian properties of triangular grid graphs are considered in (10, 11). Such proper- ties are important in applications connected with problems arising in molecular biology (protein folding) (1), in configurational statistics of polymers (3, 9), in t...

Hamiltonian Cycles in T-Graphs

Discrete & Computational Geometry, 2000

There is only one finite, 2-connected, linearly convex graph in the Archimedean triangular tiling that does not have a Hamiltonian cycle.

Hamiltonian Paths in Some Classes of Grid Graphs

Journal of Applied Mathematics, 2012

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.

Hamiltonian Paths in Two Classes of Grid Graphs

ArXiv, 2011

In this paper, we give the necessary and sufficient conditions for the existence of Hamiltonian paths in L−L-L−alphabet and C−C-C−alphabet grid graphs. We also present a linear-time algorithm for finding Hamiltonian paths in these graphs.

Hamiltonian Cycles in Linear-Convex Supergrid Graphs

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and triangular grid graphs was known to be NP-complete. In the past, we have shown that the Hamiltonian cycle problem for supergrid graphs is also NP-complete. The Hamiltonian cycle problem on supergrid graphs can be applied to control the stitching trace of computerized sewing machines. In this paper, we will study the Hamiltonian cycle property of linear-convex supergrid graphs which form a subclass of supergrid graphs. A connected graph is called kkk-connected if there are kkk vertex-disjoint paths between every pair of vertices, and is called locally connected if the neighbors of each vertex in it form a connected subgraph. In this paper, we first show that any 2-connected, linear-convex supergrid graph is locally connected. We then prove that an...

The Hamiltonian Connected Property of Some Shaped Supergrid Graphs

2017

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, parallelo...

On Hamiltonian cycles in two-triangle graphs

1987

Definition 2. Let G be a graph and e one of its edges. The number of Hamiltonian cycles containing e is denoted by cG (e) and the total number of Hamiltonian cycles occurring in G is denotedby c (G). For a given integer n^ 4, we denote by c2A (n) 9 c3CT (n)> CPT (U) the minimum possible number of Hamiltonian cycles that may occur in a Hamiltonian 2A-graph, 3-connected triangulated graph and planar triangulation on n vertices, respectively.