COMPLEXITY OF THE HAMILTONIAN CYCLE PROBLEM IN TRIANGULAR GRID GRAPHS (original) (raw)

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.

Hamiltonian properties of triangular grid graphs

Preprint 06/15, FMA, OvGU Magdeburg, 2006 (appeared later in Discrete Mathematics, 308 (24), 6166 - 6188), 2006

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, Reay and Zamfirescu showed that all 2-connected, linearly convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph D which is the linearlyconvex hull of the Star of David. We extend this result to a wider class of locally connected triangular grid graphs. Namely, we prove that all connected, locally connected triangular grid graphs (with the same exception of graph D) are hamiltonian. Moreover, we present a sufficient condition for a connected graph to be fully cycle extendable.

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.

Bend Complexity and Hamiltonian Cycles in Grid Graphs

Lecture Notes in Computer Science, 2017

Let G be an m × n rectangular grid graph. We study the problem of transforming Hamiltonian cycles on G under two basic operations we call flip and transpose. We introduce a new complexity measure, the bend complexity, for Hamiltonian cycles. Given any two Hamiltonian cycles C1 and C2 of bend complexity 1, we show that C1 can be transformed to C2 using only a linear number of flips and transposes.

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.

Logo Hamiltonian Paths in C − shaped Grid Graphs

2018

One of the well-known NP-complete problems in graph theory i s the Hamiltonian path problem; i.e., finding a simple path in the graph such that every vertex visits exactl y 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 bet ween them is equal to 1. A grid graph Gg is a finite vertex-induced subgraph of the two-dimensional integer gr id 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}, wherevx andvy arex andy coordinates of v, respectively. AC−shaped grid graph C(m, n, k, l) is a rectangular grid graphR(m, n) such that a rectangular subgraph R(k, l) is removed from it whileR(m, n) andR(k, l) have exactly one border side in common, where k, l ≥ 1 andm, n > 1 (see Fig. 1(c)). ...

Reconfiguration of Hamiltonian cycles and paths in grid graphs

2020

A grid graph is a finite embedded subgraph of the infinite integer grid. A solid grid graph is a grid graph without holes, i.e., each bounded face of the graph is a unit square. The reconfiguration problem for Hamiltonian cycle or path in a sold grid graph G asks the following question: given two Hamiltonian cycles (or paths) of G, can we transform one cycle (or path) to the other using some “operation” such that we get a Hamiltonian cycle (or path) of G in the intermediate steps (i.e., after each application of the operation)? In this thesis, we investigate reconfiguration problems for Hamiltonian cycles and paths in the context of two types of solid graphs: rectangular grid graphs, which have a rectangular outer boundary, and L-shaped grid graphs, which have a single reflex corner on the outer boundary, under three operations we define, flip, transpose and switch, that are local in the grid. Reconfiguration of Hamiltonian cycles and paths in embedded grid graphs has potential appl...

Enumeration of Hamiltonian Cycles in Some Grid Graphs

In polymer science, Hamiltonian paths and Hamiltonian circuits can serve as excellent simple models for dense packed globular proteins. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate thermodynamics of protein folding. Hamiltonian circuits are a mathematical idealization of polymer melts, too. The number of Hamiltonian cycles on a graph corresponds to the entropy of a polymer system. In this paper, we present new characterizations of the Hamiltonian cycles in a labeled rectangular grid graph P m × P n and in a labeled thin cylinder grid graph C m ×P n . We proved that for any fixed m, the numbers of Hamiltonian cycles in these grid graphs, as sequences with counter n, are determined by linear recurrences. The computational method outlined here for finding these difference equations together with the initial terms of the sequences has been implemented. The generating functions of the sequences are given explicitly for some values of m. The obtained data are consistent with data obtained in the works by Kloczkowski and Jernigan, and Schmalz et al.

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.