Squeeze-free Hamiltonian Paths in Grid Graphs

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.

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.

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.

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...

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...

The Journal of Supercomputing, 2017

Manoussakis, Y., A linear-time algorithm for finding Hamiltonian cycles in tournaments, Discrete Applied Mathematics 36 (1992) 199-201. We give a simple algorithm to transform a Hamiltonian path in a Hamiltonian cycle, if one exists, in a tournament T of order n. Our algorithm is linear in the number of arcs, i.e., of complexity O(m) = O(n') and when combined with the O(n log n) algorithm of [2] to find a Hamiltonian path in T, it yields an O(n') algorithm for searching a Hamiltonian cycle in a tournament. Up to now, algorithms for searching Hamiltonian cycles in tournaments were of order 0(n3) [3], or O(n'log I?) [S].

Ars Combinatoria -Waterloo then Winnipeg-

The link length of a walk in a multidimensional grid is the number of straight line segments constituting the walk. Alternatively, it is the number of turns that a mobile unit needs to perform in traversing the walk. A rectilinear walk consists of straight line segments which are parallel to the main axis. We wish to construct rectilinear walks with minimal link length traversing grids. If G denotes the multidimensional grid, let s(G) be the minimal link length of a rectilinear walk traversing all the vertices of G. In this paper we develop an asymptotically optimal algorithm for constructing rectilinear walks traversing all the vertices of complete multidimensional grids and analyze the worst-case behavior of s(G), when G is a multidimensional grid.

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.

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.

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.