Hamiltonian Paths in Some Classes of 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...

Lecture Notes in Computer Science, 2022

A simple s, t path P in a rectangular grid graph G is a Hamiltonian path from the top-left corner s to the bottom-right corner t such that each internal subpath of P with both endpoints a and b on the boundary of G has the minimum number of bends needed to travel from a to b (i.e., 0, 1, or 2 bends, depending on whether a and b are on opposite, adjacent, or the same side of the bounding rectangle). Here, we show that P can be reconfigured to any other simple s, t path of G by switching 2 × 2 squares, where at most 5|G|/4 such operations are required. Furthermore, each square-switch is done in O(1) time and keeps the resulting path in the same family of simple s, t paths. Our reconfiguration result proves that the Hamiltonian path graph G for simple s, t paths is connected and has diameter at most 5|G|/4 which is asymptotically tight.

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...

2015

Our studies are related to a special class of FASS-curves, which can be described in a node-rewriting Lindenmayer-system. These ortho-tile (or diagonal) type recursive curves inducing Hamiltonian paths. We define a special directed graph on a rectangular grid, and we enumerate all Hamiltonian paths on this graph. Our formulas are strongly related to both the Fibonacci numbers and the domino tilings of chessboards. The constructability of the regular 17-gon with straightedge and compass is also related.

2006

We study the Hamiltonian Cycle problem in graphs induced by subsets of the vertices of the tiling of the plane with equilateral triangles. By analogy with grid graphs we call such graphs triangular grid graphs. Following the analogy, we define the class of solid triangular grid graphs. We prove that the Hamiltonian Cycle problem is NPcomplete for triangular grid graphs. We show that with the exception of the "Star of David", a solid triangular grid graph without cut vertices is always Hamiltonian.

2015

Motivated by multi-robot construction systems, we introduce the problem of nding squeeze-free Hamiltonian paths in grid graphs. A Hamiltonian path is squeeze-free if it does not pass between two previously visited vertices lying on opposite sides. We determine necessary and sucient conditions for the existence of squeeze-free Hamiltonian paths in staircase grid graphs. Our proofs are constructive and lead to linear time algorithms for determining such paths, provided that they exist.

2019

The Hamiltonian path problem on general graphs is well-known to be NP-complete. In the past, we have proved it to be also NP-complete for supergrid graphs. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices in it. Determining whether a supergrid graph is Hamiltonian connected is clear to be NPcomplete. Recently, we proved the Hamiltonian connectivity of some special supergrid graphs, including rectangular, triangular, parallelogram, and trapezoid. In this paper, we will study the Hamiltonian connectivity of alphabet supergrid graphs. There are 26 types of alphabet supergrid graphs in which every capital letter is represented by a type of alphabet supergrid graphs. We will prove L-, C-, F-, E-, N-, and Y-alphabet supergrid graphs to be Hamiltonian connected. The Hamiltonian connectivity of the other alphabet supergrid graphs can be verified similarly. The Hamiltonian connected property of alphabet supergrid graphs can be appli...

2019

For a (p, q)-graph G, if the vertices of G can be arranged in a sequence v1, v2, . . . , vp such that for each i = 1, 2, . . . , p − 1, the distance from vi to vi+1 equal to k, then the sequence is called an AL(k)-step traversal. Furthermore, if d(vp, v1) = k, the sequence v1, v2, . . . , vp, v1 is called a k-step Hamiltonian tour and G is k-step Hamiltonian. In this paper we completely determine which rectangular grid graphs are 3-step Hamiltonian and show that the torus graph Cm×Cn is 3-step Hamiltonian for all m ≥ 3, n ≥ 5.

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.

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.