Rectangular Grid Graph Research Articles

The hamiltonian path graph is connected for simple s, t paths in rectangular grid graphs

The hamiltonian path graph is connected for simple s, t paths in rectangular grid graphs

Hamiltonicity of rectangular grid graphs (meshes) with an L-shaped hole

Hamiltonicity of rectangular grid graphs (meshes) with an L-shaped hole

An Algorithm for the Numbers of Homomorphisms from Paths to Rectangular Grid Graphs

Let G and H be graphs. A mapping f from the vertices of G to the vertices of H is known as a homomorphism from G to H if, for every pair of adjacent vertices x and y in G, the vertices f(x) and f(y) are adjacent in H. A rectangular grid graph is the Cartesian product of two path graphs. In this paper, we provide a formula to determine the number of homomorphisms from paths to rectangular grid graphs. This formula gives the solution to the problem concerning the number of walks in the rectangular grid graphs.

Reconfiguration of Hamiltonian Cycles in Rectangular Grid Graphs

We study reconfiguration of simple Hamiltonian cycles in a rectangular grid graph [Formula: see text], where the Hamiltonian cycle in each step of the reconfiguration connects every internal vertex of [Formula: see text] to a boundary vertex by a single straight line segment. We introduce two operations, flip and transpose, which are local to the grid. We show that any simple cycle of [Formula: see text] can be reconfigured to any other simple cycle of [Formula: see text] using [Formula: see text] flip and transpose operations. Our result proves that the simple Hamiltonian cycle graph [Formula: see text] is connected with respect to those two operations and has diameter [Formula: see text].

1-Complex $s,t$ Hamiltonian Paths: Structure and Reconfiguration in Rectangular Grids

We give a complete structure theorem for $1$-complex $s,t$ Hamiltonian paths in rectangular grid graphs. We use the structure theorem to design an algorithm to reconfigure one such path into any other in linear time, making a linear number of switch operations in grid cells.

The Longest Path Problem in Odd-Sized O-Shaped Grid Graphs

One of the well-known NP-hard optimization problems in graph theory is finding the longest path in a graph. This problem remains NP-hard for general grid graphs, and its complexity is open for grid graphs that have a limited number of holes. In this paper, we study this problem for odd-sized [Formula: see text]-shaped grid graphs, i.e. a rectangular grid graph with a rectangular hole. We show that this problem can be solved in linear time.

Two counterparts of the TFK formula for cylinder graphs

The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph — due independently to Temperley and Fisher, and Kasteleyn, who found the TFK formula — consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In this paper we present a new method, which relies on the Cauchy-Binet theorem, and use it to solve the two dimensional dimer problem for cylinder graphs on the square and on the hexagonal lattice.We provide explicit product formulas for these two kinds of cylinder graphs. One advantage of our formula for the square lattice compared to the TFK formula is that ours has a linear number of factors, while the number of factors in the former is quadratic. Our result for the hexagonal lattice yields a formula for the number of periodic stepped surfaces that fit in an infinite tube of given cross-section. This can be regarded as a counterpart of MacMahon's boxed plane partition theorem.

A linear-time algorithm for finding Hamiltonian cycles in rectangular grid graphs with two rectangular holes

The Hamiltonian cycle problem is one of the most important problems in graph theory that has many applications. This problem is NP-complete for general grid graphs. For solid grid graphs, there are polynomial-time algorithms. Existence of polynomial-time algorithms for grid graphs with few holes has been asked in the literature. In this paper, we give a linear-time algorithm for rectangular grid graphs with two rectangular holes. This extends the previous result for rectangular grid graphs with one rectangular hole. We first present the forbidden conditions in which there is no Hamiltonian cycle for any grid graphs, including rectangular grid graphs with rectangular holes. We then show that if these forbidden conditions do not hold, then there exists a Hamiltonian cycle. The proof is constructive, therefore, it gives an algorithm. An application of the problem is in off-line robot exploration.

The structure of the 2-factor transfer digraph common for rectangular, thick cylinder and Moebius strip grid graphs

In this paper, we prove that all but one of the components of the transfer digraph D? m needed for the enumeration of 2-factors in the rectangular, thick cylinder and Moebius strip grid graphs of the fixed width m (m ? N) are bipartite digraphs and that their orders could be expressed in term of binomial coefficients. In addition, we prove that the set of vertices of each component consists of all the binary m-words for which the difference of numbers of zeros in odd and even positions is constant.

Encoding Vertices in Rectangular Grid Graphs with Eliminating Errors

An undirected graph G = (V,E) where V is a set of vertices and E =V ×V is the set of pair of adjacent edges or in other words it is the set of edges. In theory, a graph can be a model of a message delivery in a network. We assume that the computer network has a particular shape which we call as a rectangular grid and there is a computer on each vertex in the graph. Each vertex v∈V is labelled by a subset of universal set U that models the header of a message sent between two distinct computers in G. We present a way to encode routes in the graph G by encoding all distinct vertices u,v ∈V in the routes. We aim that these codes prevent errors denoted by false positives, therefore, results in a more efficient use of network resources.

Finding Hamiltonian cycles of truncated rectangular grid graphs in linear time

The Hamiltonian cycle problem is an important problem in graph theory. For solid grid graphs, an O(n4)-time algorithm has been given. In this paper, we solve the problem for a special class of solid grid graphs, i.e. truncated rectangular grid graphs, in linear time.

Application of Doubly Connected Dominating Sets to Safe Rectangular Smart Grids

Smart grids, together with the Internet of Things, are considered to be the future of the electric energy world. This is possible through a two-way communication between nodes of the grids and computer processing. It is necessary that the communication is easy and safe, and the distance between a point of demand and supply is short, to reduce the electricity loss. All these requirements should be met at the lowest possible cost. In this paper, we study a two-dimensional rectangular grid graph which is considered to be a model of a smart grid; nodes of the graph represent points and devices of the smart grid, while links represent possible ways of communication and energy transfer. We consider the problem of choosing the lowest possible number of locations (nodes, points) of the grid which could serve as energy sources (or a source of different resources) to other nodes in such a way that we ensure reduction in electricity loss and provide safe communication and resistance to failures and increases in energy demand.Therefore, we study minimum doubly connected dominating sets in grid graphs. We show that the proposed solutions are the best possible in terms of the number of source points for the case of narrow grid graphs and we give upper and lower bounds for the case of wide grid graphs.

Routing by matching on convex pieces of grid graphs

The routing number is a graph invariant introduced by Alon, Chung, and Graham in 1994, and it has been studied for trees and other classes of graphs such as hypercubes. It gives the minimum number of routing steps needed to sort a set of distinct tokens, placed one on each vertex, where each routing step swaps a set of disjoint pairs of adjacent tokens. Our main theorem generalizes the known estimate that a rectangular grid graph R with width w(R) and height h(R) satisfies rt(R)∈O(w(R)+h(R)). We show that for the subgraph P of the infinite square lattice enclosed by any convex polygon, we have rt(P)∈O(w(P)+h(P)).

NP-Completeness and One Polynomial Subclass of the Two-Step Graph Colouring Problem

This paper considers the two-step colouring problem for an undirected connected graph. The problem is about colouring the graph in a given number of colours so that no pair of vertices at a distance of 1 or 2 between each other has the same colour. Also the corresponding recognition problem is set. The problem is closely related to the classical graph colouring problem. In the article, the polynomial reduction of the problems to one another is analyzed and proved. In particular, this allows us to prove the NP-completeness of the two-step colouring problem. Also we specify some of its properties. The two-step colouring problem as applied to rectangular grid graphs is considered separately. The maximum vertex degree in these graphs ranges from 0 to 4. The function of two-vertex colouring in the minimum possible number of colours was defined and proved for each case. The resulting functions are drawn so that each vertex is coloured independently from others. If the vertices are examined in a sequence, the time for colouring a rectangular grid graph will be polynomial.

Maker–Breaker percolation games I: crossing grids

AbstractMotivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n.Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ.If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m.Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete (“witnesses exist but are hard to find”), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily “cancels” the effect of another clue in the same region, makes path finding Σ2-complete (“witnesses do not exist”), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving “subset Hamiltonian path” for terminals on the boundary of an embedded planar graph in polynomial time.

Construction of $\sigma ^+$-game solutions on a rectangular grid

The $\sigma ^+$-game is played on a directed graph where the vertices (lights) are assigned an off or on state. The player can press any light, thereby toggling it along with its neighbors. The objective of the game is to turn all lights off given an initial configuration. Our particular interest are rectangular grid graphs with all lights initially on. We explore different classes of grid sizes and dimensions with unique solutions, and we construct the solutions to grids of size $(2^n-1)\times (2^n-1)$ with all lights initially on.

NP-completeness and One Polynomial Subclass of the Two-Step Graph Colouring Problem

In this paper, we study the two-step colouring problem for an undirected connected graph. It is required to colour the graph in a given number of colours in a way, when no pair of vertices has the same colour, if these vertices are at a distance of 1 or 2 between each other. Also the corresponding recognition problem is set. The problem is closely related to the classical graph colouring problem. In the article, we study and prove the polynomial reduction of the problems to each other. So it allows us to prove NP-completeness of the problem of two-step colouring. Also we specify some of its properties. Special interest is paid to the problem of two-step colouring in application to rectangular grid graphs. The maximum vertex degree in such a graph is between 0 and 4. For each case, we elaborate and prove the function of two-vertex colouring in the minimum possible number of colours. The functions allow each vertex to be coloured independently from others. If vertices are examined in a sequence, colouring time is polynomial for a rectangular grid graph.

Linear-time algorithms for finding Hamiltonian and longest [formula omitted]-paths in [formula omitted]-shaped grid graphs

The longest and Hamiltonian path problems are well-known NP-hard problems in graph theory. Despite many applications of these problems, they are still open for many classes of graphs, including solid grid graphs and grid graphs with some holes. We consider the longest and Hamiltonian (s,t)-path problems in C-shaped grid graphs. A (s,t)-path is a path between two given vertices s and t of the graph. A C-shaped grid graph is a rectangular grid graph such that a rectangular grid subgraph is removed from it to make a C-liked shape. In this paper, we first give the necessary conditions for the existence of Hamiltonian cycles and Hamiltonian (s,t)-paths in such graphs. Then by given a linear-time algorithm for finding Hamiltonian cycles and Hamiltonian (s,t)-paths, we show that these necessary conditions are also sufficient. Finally, we give a linear-time algorithm for finding the longest (s,t)-path in these graphs.

On Coloring Rectangular and Diagonal Grid Graphs for Multipatterning and DSA Lithography

Rectangular grid graph (RGG) and diagonal grid graph (DGG) are induced subgraphs of a rectangular or diagonal grid, respectively. Their $k$ -coloring problem has direct applications in printing contact/via layouts by multipatterning lithography (MPL). However, the problem in general is computationally difficult for $k>2$ , while it remains an open question on grid graphs due to their regularity and sparsity. On the other hand, directed self-assembly (DSA) technique can work with MPL to optimize the graph by grouping neighboring vertices such that $k$ can be reduced, but the problem of deploying the grouping for coloring is even more intractable. In this paper, we study both of the $k$ -coloring problems, with and without DSA grouping, on RGG and DGG. Without grouping, a complete $k$ -coloring analysis is conducted and particularly the NP-completeness of 3-coloring on a diagonal grid is proved. When considering grouping, we present a 3-coloring solution and prove the NP-completeness to solve the problem of $k=2$ .