Grid Graphs Research Articles

Magnetic square lattice with vertex coupling of a preferred orientation

We analyze a square lattice graph in a magnetic field assuming that the vertex coupling is of a particular type violating the time reversal invariance. Calculating the spectrum numerically for rational values of the flux per plaquette we show how the two effects compete; at the high energies it is the magnetic field which dominates restoring asymptotically the familiar Hofstadter’s butterfly pattern.

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.

Independence complexes of (n × 4) and (n × 5)-grid graphs

We determine the homotopy types of the independence complexes of (n×4) and (n×5)-square grid graphs. In fact, they are homotopy equivalent to wedges of spheres.

Formation of Non-Perfect Maze Using Prim’s Algorithm

Maze is a place that has many paths with tortuous paths that are misleading and full of dead ends and can be viewed as a grid graph. A non-perfect maze is a maze that has a cycle. This research produces an algorithm that can form a non-perfect maze with a size of m×n which has two types of bias. The first bias is the composition of the percentage of horizontal and vertical partitions. The second bias is the percentage of the number of cycles. The algorithm created in this study was generated by modifying Prim’s algorithm and the use of Fisher-Yates algorithm which is used in random selection in Prim’s algorithm. The non-perfect maze algorithm begins with the calculation of the parameter values of the two types of bias and continues with forming a perfect maze and ends with forming a non-perfect maze. The algorithm that has been designed can form a non-perfect maze with a complexity of O(|E|^2), where E is the set of edges of an m×n grid graph. Flash-based application development is also carried out in order to implement algorithms to obtain a non-perfect maze. The non-perfect maze is produced in a two-dimensional visual form in the form of an image along with its corresponding grid graph. The application is capable of displaying up to the first 20 solutions of the biased maze.

Networks of splicing processors: simulations between topologies

Networks of splicing processors are one of the theoretical computational models that take inspiration from nature to efficiently solve problems that our current computational knowledge is not able to. One of the issues restricting/hindering is practical implementation is the arbitrariness of the underlying graph, since our computational systems usually conform to a predefined topology. We propose simulations of networks of splicing processors having arbitrary underlying graphs by networks whose underlying graphs are of a predefined topology: complete, star, and grid graphs. We show that all of these simulations are time efficient in the meaning that they preserve the time complexity of the original network: each computational step in that network is simulated by a fixed number of computational steps in the new topologic networks. Moreover, these simulations do not modify the order of magnitude of the network size.

On the computational complexity of Roman$$\{2\}$$-domination in grid graphs

On the computational complexity of Roman$$\{2\}$$-domination in grid graphs

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.

Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

Secure connected domination and secure total domination in unit disk graphs and rectangle graphs

Given a graph G with vertex set V, a secure connected (resp. total) dominating set of G is a connected (resp. total) dominating set S⊆V with the property that for each u∈V∖S, there exists v∈S adjacent to u such that (S∪{u})∖{v} is a connected (resp. total) dominating set of G. The minimum secure connected dominating set (or, for short, MSCDS) (resp. minimum secure total dominating set (or, for short, MSTDS)) problem is to find an MSCDS (resp. MSTDS) in a given graph.In this paper, we initiate to consider complexity and algorithmic aspects of the MSCDS problem and the MSTDS problem in unit disk graphs and rectangle graphs. Firstly, we show that the decision version of the MSCDS problem is NP-complete in unit square graphs and unit disk graphs. Then we show that the decision version of the MSTDS problem is NP-complete even in grid graphs (a subclass of unit square graphs and unit disk graphs). Secondly, we give linear-time constant-approximation algorithms for the two problems in unit square graphs, unit disk graphs and unit-height rectangle graphs. Thirdly, we propose a PTAS for the MSTDS problem in unit square graphs and unit disk graphs. Finally, we show that the two problems in proper rectangle graphs are APX-hard. Further we give an explicit lower bound 1.00147 on efficient approximability for the two problems in proper rectangle graphs unless P=NP.

Restrained domination and its variants in extended supergrid graphs

Domination problem is a well-known NP-complete problem for general graphs. In this paper, we will study its three variants, including restrained, independent restrained, and restrained-step dominations. A restrained dominating set of a graph is a set R of vertices of the graph such that every vertex of the graph is in R or adjacent to at least a vertex of R, and the subgraph induced by vertices not in R contains no isolated vertex. An independent restrained dominating set of a graph is a restrained dominating set such that its any two vertices are not adjacent. A restrained-step dominating set of a graph is a restrained dominating set such that the subgraph induced by vertices not in this set contains a perfect matching. Restrained domination, independent restrained domination, and restrained-step domination problems are to find a restrained dominating set, independent restrained dominating set, and restrained-step dominating set with the least number of vertices on a graph, respectively. In 2015, we define supergrid graphs for computing the tracks of computer embroidery sewing machine and 3D printer. In this paper, we expand it to the class of extended supergrid graphs. Extended supergrid graphs are a natural extension of supergrid graphs, and they contain grid, triangular grid, and supergrid graphs as subclasses. The complexities of the restrained related domination problems on extended supergrid graphs remain unknown. In this paper, we will prove the restrained and independent restrained domination problems on supergrid graphs is NP-complete. Using the same technique, the restrained domination problem on grid graphs can also be shown to be NP-complete. We then solve the restrained related domination problems on rectangular supergrid graphs, which form a subclass of supergrid graphs, in linear time. Finally, we provide a tight upper bound on the minimum size of (independent) restrained dominating set for rectangular triangular-supergrid graphs forming a subclass of triangular grid graphs.

Class of Crosscap Two Graphs Arising from Lattices–I

Let L be a lattice. The annihilating-ideal graph of L is a simple graph whose vertex set is the set of all nontrivial ideals of L and whose two distinct vertices I and J are adjacent if and only if I∧J=0. In this paper, crosscap two annihilating-ideal graphs of lattices with at most four atoms are characterized. These characterizations provide the classes of multipartite graphs, which are embedded in the Klein bottle.

Grid recognition: Classical and parameterized computational perspectives

Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph is hard—it was shown to be NP-hard already in 1987. In this paper, we provide several positive results in this regard in the framework of parameterized complexity. Specifically, our contribution is threefold. First, we show that the problem is FPT parameterized by k+mcc where mcc is the maximum size of a connected component of G. Second, we present a new parameterization, denoted aG, relating graph distance to geometric distance. We show that the problem is para-NP-hard parameterized by aG, but FPT parameterized by aG on trees, as well as FPT parameterized by k+aG. Third, we show that the recognition of k×r grid graphs is NP-hard on graphs of pathwidth 2 where k=3.

Independence in multiresolving sets of graphs

ABSTRACT In this work, we introduce the concept of independence in multiresolving sets and present some basic results on it. We characterize two graph families for which we prove that their multiset dimension is three. We also provide a corrected proof for the multiset dimension of the grid graph , due to Simanjuntak et al., present in the literature.

A quasi-solution on a lattice graph for an initial state reconstruction

This note considers a graph-based initial state reconstruction in image denoising. Our particular focus is on developing a quasi-model by a combination of boundary perturbation methods. The corresponding spectral quasi-solution enables one to stably approximate the original pixelated image after the standard forward–backward denoising process. Convergence estimates are derived with respect to the deterministic noise level. Some numerical examples are provided to verify the performance of the method.

A combat game model with inter-network confrontation and intra-network cooperation

Inter-network combat and intra-network cooperation among structured systems are likely to have been recurrent features of human evolutionary history; however, little research has investigated the combat mechanism between structured systems that the adversarial interactions will cause the disability of agents and agents are prone to seek cooperation with neighbors. Hence, the current study has proposed a two-network combat game model and designed the corresponding rules of how to attack, how to be disabled, how to cooperate, and how to win. First, within the framework of our model, we have simulated the combat among four common network structures-the Erdős-Rényi (ER) random network, the grid graph, the small-world network, and the scale-free network. We found that the grid network always holds the highest winning percentage, while the ER random graph is most likely to lose when combating with the other three network structures. For each structure, we have also simulated the combat between the same network structures with different generating parameters. The simulations reveal that the small-world property and heterogeneity can promote winning a combat. Besides, by broadening and deepening cooperation, we have found that broader cooperation helps defeat the opposite system on grid and scale-free networks, yet hinders it on ER and Watts-Strogatz (WS) networks, while deeper cooperation can benefit to winning except on scale-free networks. These findings inform our understanding of the effects of structure and cooperation in a combat.

Non-communicating Decentralized Multi-robot Collision Avoidance in Grid Graph Workspace based on Dueling Double Deep Q-Network

This paper designs a motion rule suitable for grid environment and proposes a multi-robot autonomous obstacle avoidance method based on deep reinforcement learning. The training and validation of the method are done under the Stage simulation platform based on ROS operating system. During the training process, the robot uses Lidar to obtain the surrounding state information and generates actions based on the state information to obtain rewards, and the robot is guided by the rewards to optimize the strategy. Based on the D3QN algorithm, a new reward function is designed, a proximity penalty is introduced to reduce the collision between robots, a distance reward is added to guide the robot to complete the task, a step reward is added to improve the efficiency of the robot to complete the task, and an illegal action penalty is added to avoid the robot to choose an illegal action; the input is 5 frames of Lidar data, and in the network structure, the agent can better learn the correlation between the data by introducing Long Short Term Memory(LSTM) layer, and introducing Convolutional Block Attention Module(CBAM), a hybrid attention mechanism to allow the robot to pay more attention to the information of the surrounding robots. By designing experiments, we demonstrate that the learned strategy can effectively guide the robot through obstacle avoidance and complete the task.

A Spanning Union of Cycles in Thin Cylinder, Torus and Klein Bottle Grid Graphs

In this paper, we propose an algorithm for obtaining the common transfer digraph Dm* for enumeration of 2-factors in graphs from the title, all of which have m·n vertices (m,n∈N,m≥2). The numerical data gathered for m≤18 reveal some matches for the numbers of 2-factors for different types of torus or Klein bottle. In the latter case, we conjecture that these numbers are invariant under twisting.

PIGNN-CFD: A physics-informed graph neural network for rapid predicting urban wind field defined on unstructured mesh

Urban wind field plays an important role in quantitative assessment of urban environment. Compared to field measurement and wind tunnel experiment, Computational Fluid Dynamics (CFD) simulation having the advantages of low cost, repeatability and reliable precision is becoming a common scheme to model flow field of one fixed urban scenario, but still faces the problems of time-consuming computation and lack of scalability for practical engineering application. This paper proposes PIGNN-CFD, a novel physics-informed graph neural network for rapid predicting urban wind field based on irregular unstructured mesh data of CFD simulation. Specifically, a CFD model employing the unsteady Reynolds-Averaged Navier-Stokes (RANS) equations with the standard k-ε turbulence model, is constructed and then numerically solved by OpenFOAM to simulate urban wind field defined on unstructured mesh. After being validated by publicly available wind tunnel test data provided by the Architectural Institute of Japan (AIJ), the proposed CFD model is employed to build the training and test sample sets of urban wind fields by simulating the wind blowing through various randomly generated small-scale urban scenes. A novel physics-informed graph neural network, both approximating the training data and automatically satisfying the RANS equations, is designed and trained to perform wind field inference on unstructured mesh graph, and then scaled up to predict wind fields of arbitrary large-scale urban scenes. The predicted wind field results of two urban environments at different scales show that the well-generalized PIGNN-CFD model runs 1–2 orders of magnitude faster than the CFD model on which it is trained, while obtaining the consistent computational accuracy.

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.