Grid Graphs Research Articles

Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs

In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N, r>1 and 0<α<N, −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=∑y∈KN,y≠x|u(y)|rd(x,y)N−α|u|r−2u, where Δν is the discrete ν-Laplacian on graphs, and ν∈{p.q}, V(x) is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.

Expected 1.x Makespan-Optimal Multi-Agent Path Finding on Grid Graphs in Low Polynomial Time

Multi-Agent Path Finding (MAPF) is NP-hard to solve optimally, even on graphs, suggesting no polynomial-time algorithms can compute exact optimal solutions for them. This raises a natural question: How optimal can polynomial-time algorithms reach? Whereas algorithms for computing constant-factor optimal solutions have been developed, the constant factor is generally very large, limiting their application potential. In this work, among other breakthroughs, we propose the first low-polynomial-time MAPF algorithms delivering 1-1.5 (resp., 1-1.67) asymptotic makespan optimality guarantees for 2D (resp., 3D) grids for random instances at a very high 1/3 agent density, with high probability. Moreover, when regularly distributed obstacles are introduced, our methods experience no performance degradation. These methods generalize to support 100% agent density. Regardless of the dimensionality and density, our high-quality methods are enabled by a unique hierarchical integration of two key building blocks. At the higher level, we apply the labeled Grid Rearrangement Algorithm (GRA), capable of performing efficient reconfiguration on grids through row/column shuffles. At the lower level, we devise novel methods that efficiently simulate row/column shuffles returned by GRA. Our implementations of GRA-based algorithms are highly effective in extensive numerical evaluations, demonstrating excellent scalability compared to other SOTA methods. For example, in 3D settings, GRA-based algorithms readily scale to grids with over 370,000 vertices and over 120,000 agents and consistently achieve conservative makespan optimality approaching 1.5, as predicted by our theoretical analysis.

Existence and Multiplicity of Solutions for the Logarithmic Schrödinger Equation with a Potential on Lattice Graphs

Existence and Multiplicity of Solutions for the Logarithmic Schrödinger Equation with a Potential on Lattice Graphs

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

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

Benchmarking Multipartite Entanglement Generation with Graph States

AbstractAs quantum computing technology slowly matures and the number of available qubits on a QPU gradually increases, interest in assessing the capabilities of quantum computing hardware in a scalable manner is growing. One of the key properties for quantum computing is the ability to generate multipartite entangled states. In this study, aspects of benchmarking entanglement generation capabilities of noisy intermediate‐scale quantum (NISQ) devices are discussed based on the preparation of graph states and the verification of entanglement in the prepared states. Thereby, entanglement witnesses that are specifically suited for a scalable experiment design are used. This choice of entanglement witnesses can detect A) bipartite entanglement and B) genuine multipartite entanglement for graph states with constant two measurement settings if the prepared graph state is based on a two‐colorable graph, e.g., a square grid graph or one of its subgraphs. With this, it is experimentally verified that a bipartite entangled state comprising all qubits can be prepared on a 127‐qubit IBM Quantum superconducting QPU, and genuine multipartite entanglement can be detected for states of up to 23 qubits with quantum readout error mitigation.

Efficient Domination in Grid Graphs

Let G = ( V , E ) be a graph. A subset S ⊆ V of vertices is an \textit{efficient dominating set} if every vertex v ∈ V is adjacent to exactly one vertex in S , where a vertex u ∈ S is considered to be adjacent to itself. Efficient domination is highly desirable in many real-world applications, and yet, in general, graphs are often not efficient. It is of value, therefore, to determine optimum ways in which inefficient graphs can be changed in order to make them efficient. It is well known, for example, that almost no m × n grid graphs have efficient dominating sets. In this paper, we consider the minimum number of vertices that can be removed from an m × n grid graph so that the remaining graph has an efficient dominating set.

Computing the Edge Irregularity Strength of Some Classes of Grid Graphs

Let G be a simple graph. A function ϕ:V(G)→{1,2,…,k} a vertex k-labeling which assigns labels to the vertices of G. For any edge xy in G, we define the weight of this edge as wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is termed as an edge irregular k -labeling of G. The smallest possible value of k for which the graph G possesses an edge irregular k-labeling is denoted as the edge irregularity strength of G and is represented as es(G). In this paper, we investigate the edge irregular k-labeling of some classes of grid graphs, namely rhombic graph Rmn, triangular graph Lmn and octagonal graph Omn. As by-product, we obtain their precise value of edge irregularity strength.

Pink-noise dynamics in an evolutionary game on a regular graph.

We consider a multiplayer prisoner's dilemma game on a square lattice and regular graphs based on the pairwise-Fermi update rule, and we obtain heatmaps of the fraction of cooperators and the correlation of neighboring pairs. In the heatmap, we find a mixed region where cooperators and defectors coexist, and the correlation between neighbors is significantly enhanced. Moreover, we observe pink-noise behavior in the mixed region, where the power spectrum can be fitted by a power-law function of frequency. We also find that the pink-noise behavior can be reproduced in a simple random-walk model. In particular, we propose a modified random-walk model which can reproduce not only the pink-noise behavior but also the deviation from it observed in a low-frequency region.

A Non-Uniform Grid Graph Convolutional Network for Sea Surface Temperature Prediction

Sea surface temperature (SST) is an important factor in the marine environment and has significant impacts on climate, ecology, and maritime activities. Most existing SST prediction methods consider the ocean as a uniform field and use a uniform grid to predict SST. However, the marine environment is a complex system, and factors such as solar radiation, differences in land and sea thermal properties, and ocean circulation lead to uneven spatial distributions of SSTs. We propose a non-uniform grid construction method based on an SST spatial gradient to encode SST data, as well as a Non-uniform Grid Graph Convolutional Network (NGGCN) model. The NGGCN consists of two spatiotemporal modules, each of which extracts spatial features from the GCN module, captures temporal correlations through the GRU module, and performs feature restoration and output results through the fully connected module. We selected data from the Yellow Sea and Bohai Sea to validate the effectiveness of the NGGCN in predicting SST at different time scales and prediction steps. The results indicate that our model shows a significant improvement in prediction performance compared to other models.

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

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

The equitable non-split domination number of graphs

Finding a group of dominant servers is a necessary step towards optimising service provisioning to all clients in the complex world of server-client networks. This crucial decision rests on the non-split dominance number in the graph architecture. We present the idea of an equitable non-split domination number in graphs to address the possible problem of unequal task distribution among servers. When the vertices of a dominating set, Dens, have the same colour in an equitable colouring scheme, the set is said to be equitable. The induced subgraph ⟨V −D⟩ also maintains its connectivity. In particular, the application of equitable non-split domination numbers to graphs like the Triangular Grid, Antiprism, Wheel, and Complete Graphs is explored in this study.

On the Complexity of Target Set Selection in Simple Geometric Networks

We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.

Total (restrained) domination in unit disk graphs

The minimum total domination problem and the minimum total restrained domination problem are classical combinatorial optimization problems. In this paper, we first show that the decision versions of both the minimum total domination problem and the minimum total restrained domination problem are NP-complete for grid graphs (a subclass of unit disk graphs), thereby strengthening the result on the total domination decision problem in unit disk graphs established by Jena et al. (2021). Then, we present a linear-time 6.387-approximation algorithm for the minimum total domination problem in unit disk graphs, improving upon the previous algorithm with a ratio of 8 by Jena et al. (2021). Finally, we propose a linear-time 10.387-approximation algorithm and a PTAS for the minimum total restrained domination problem in unit disk graphs.

Face irregular evaluations of family of grids

Let G = ( V , E , F ) be a connected plane graph. Let ρ : α ( V ) ∪ β ( E ) ∪ γ ( F ) → { 1 , 2 , … , k } be a k-labeling where α , β , γ ∈ { 0 , 1 } and ( α , β , γ ) ≠ ( 0 , 0 , 0 ) . The weight of a face f under ρ is defined as Wt ( f ) = α ∑ v ∼ f ρ ( v ) + β ∑ e ∼ f ρ ( e ) + γ ρ ( f ) . Then ρ is called a face irregular labeling of type ( α , β , γ ) if Wt ( f ) ≠ Wt ( g ) for any two faces f ≠ g . The face irregular strength of G is the smallest integer k such that G admits a face irreg ular k-labeling. In this paper we determine the face irregular strength of the grid graph P n + 1 □ P m + 1 under a labeling of type ( α , β , γ ) .

Rearrangement inequalities on the lattice graph

AbstractThe Polya–Szegő inequality in states that, given a nonnegative function , its spherically symmetric decreasing rearrangement is ‘smoother’ in the sense of for all . We study analogues on the lattice grid graph . The spiral rearrangement is known to satisfy the Polya–Szegő inequality for , the Wang‐Wang rearrangement satisfies it for and no rearrangement can satisfy it for . We develop a robust approach to show that both these rearrangements satisfy the Polya–Szegő inequality up to a constant for all . In particular, the Wang‐Wang rearrangement satisfies for all . We also show the existence of (many) rearrangements on such that for all .

A Heuristic Route Planning Scheme in Scalable VANET-Oriented Evolving Grid Graph (S-VoEGG)

A Heuristic Route Planning Scheme in Scalable VANET-Oriented Evolving Grid Graph (S-VoEGG)

A New Lower Bound for the Domination Number of Complete Cylindrical Grid Graphs

We use a dynamic programming algorithm to establish a lower bound on the domination number of complete grid graphs of the form C n □ P m , that is, the Cartesian product of a cycle C n and a path P m , for m and n sufficiently large.

The Domination Number of C n □ P m for n ≡ 2 ( mod 5 )

We use a dynamic programming algorithm to establish a new lower bound on the domination number of complete cylindrical grid graphs of the form C n □ P m , that is, the Cartesian product of a path and a cycle, when n ≡ 2 ( mod 5 ) , and we establish a new upper bound equal to the lower bound, thus computing the exact domination number for these graphs.

What Goes Around Comes Around: Covering Tours and Cycle Covers with Turn Costs

We investigate several geometric problems of finding tours and cycle covers with minimum turn cost, which have been studied in the past, with complexity, approximation results, and open problems dating back to work by Arkin et al. in 2001. Many new practical applications have spawned variants: For full coverage, all points have to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring a penalty. We show that finding a minimum-turn (full) cycle cover is NP-hard even in 2-dimensional grid graphs, solving the long-standing open Problem 53 in The Open Problems Project edited by Demaine, Mitchell and O’Rourke. We also prove NP-hardness of finding a subset cycle cover of minimum turn cost in thin grid graphs, for which Arkin et al. gave a polynomial-time algorithm for full coverage; this shows that their boundary techniques cannot be applied to compute exact solutions for subset and penalty variants. We also provide a number of positive results. In particular, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants for grid-based instances, based on LP/IP techniques. These geometric versions allow many possible edge directions (and thus, turn angles, such as in hexagonal grids or higher-dimensional variants); our approximation factors improve the combinatorial ones of Arkin et al.