Undirected Graph Research Articles

Hardness of Approximate Diameter: Now for Undirected Graphs

Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open whether a better approximation is possible in near-linear time. A series of papers on fine-grained complexity have led to strong hardness results for diameter in directed graphs, culminating in a recent tradeoff curve independently discovered by [Li, STOC’21] and [Dalirrooyfard and Wein, STOC’21], showing that under the Strong Exponential Time Hypothesis (SETH), for any integer k ≥ 2 and δ > 0, a \(2-\frac{1}{k}-\delta \) approximation for diameter in directed m -edge graphs requires m 1 + 1/( k − 1) − o (1) time. In particular, the simple linear time 2-approximation algorithm is optimal for directed graphs. In this paper we prove that the same tradeoff lower bound curve is possible for undirected graphs as well, extending results of [Roditty and Vassilevska W., STOC’13], [Li’20] and [Bonnet, ICALP’21] who proved the first few cases of the curve, k = 2, 3 and 4, respectively. Our result shows in particular that the simple linear time 2-approximation algorithm is also optimal for undirected graphs. To obtain our result, we extract the core ideas in known reductions and introduce a unification and generalization that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation.

Some properties of a graph associated with a non-quasi-local weakly factorial domain and a supergraph of it

Let [Formula: see text] be a weakly factorial domain which admits at least two maximal ideals. We use [Formula: see text] to denote the set of all primary elements of [Formula: see text] and denote [Formula: see text] by [Formula: see text]. Let [Formula: see text]. We denote the subset of [Formula: see text] consisting of all [Formula: see text] such that [Formula: see text] does not belong to the Jacobson radical of [Formula: see text] by [Formula: see text]. With [Formula: see text], in this paper, we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In Secs. 2 and 3 of this paper, we discuss results regarding the connectedness, the girth, and the clique number of [Formula: see text] and study the interplay between graph properties of [Formula: see text] and the properties of [Formula: see text]. In Secs. 4 and 5 of this paper, we consider a supergraph of [Formula: see text], denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and study some graph properties of [Formula: see text].

Degree-Constrained Steiner Problem in Graphs with Capacity Constraints

The degree-constrained Steiner problem in graphs is well known in the literature. In an undirected graph, positive integer degree bounds are associated with nodes and positive costs with the edges. The goal is to find the minimum cost tree spanning a given node set while respecting the degree bounds. As it is known, finding a tree satisfying the constraints is not always possible. The problem differs when the nodes can participate multiple times in the coverage and the constraints represent a limited degree (a capacity) for each occurrence of the nodes. The optimum corresponds to a graph-related structure, i.e., to a hierarchy. Finding the solution to this particular Steiner problem is NP-hard. We investigate the conditions of its existence and its exact computation. The gain of the hierarchies is demonstrated by solving ILPs to compute hierarchies and trees. The advantages of the spanning hierarchies are conclusive: (1) spanning hierarchies can be found in some cases where spanning trees matching the degree constraints do not exist; (2) the cost of the hierarchy can be lower even if the Steiner tree satisfying the constraints exists.

Metric and strong metric dimension in intersection power graphs of finite groups

Let G be a finite group and let e be its identity element. The intersection power graph of G is the undirected graph with vertex set G, in which two distinct vertices x and y are adjacent if either 〈 x 〉 ∩ 〈 y 〉 ≠ { e } or one of x and y is e. In this paper, we first compute the strong metric dimension of the intersection power graph of a cyclic group, a dihedral group, and a generalized quaternion group. We also characterize all finite groups G whose intersection power graph has strong metric dimension | G | − 2 . Then, we give the sharp upper and lower bounds for the metric dimension of the intersection power graph of a finite group. As applications, we obtain the metric dimension of the intersection power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

One-to-one Disjoint-path Covers of Leaf-sort Graphs

Parallel paths of interconnection networks have attracted much attention in parallel computing systems, they are studied in terms of disjoint paths in an undirected graph [Formula: see text]. A [Formula: see text]-disjoint path cover (k-DPC for short) of a graph [Formula: see text] is a set of [Formula: see text] (internally) disjoint paths that altogether cover every vertice of the graph. A bipartite graph [Formula: see text] is one-to-one bi-k-DPC if there is a k-DPC between any pair of vertices in different partite sets. A bipartite graph is hamiltonian laceable if there exists a hamiltonian path between any pair of vertices in different partite sets. The [Formula: see text]-dimensional leaf-sort graph [Formula: see text] has many good properties, including bipartite, symmetry, vertex-transitive. In this paper, we prove that [Formula: see text] has one-to-one bi-[Formula: see text]-DPC between any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] in different partite sets, and [Formula: see text] is hamiltonian laceable.

Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

ABSTRACTIn this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.

The undirected annihilating-ideal graphs over non-commutative rings

The undirected annihilating-ideal graphs over non-commutative rings

Continuous fiber printing path generation based on Euler diagram division

Continuous fiber printed structures are widely used in aerospace applications because of their excellent performance and high design freedom. Existing path planning methods optimize the continuous fiber printing direction by stress lines or discrete angles, but these methods are far from perfect printing paths and lack planning for connectivity relations. In this study, a planning method for continuous fiber printing paths based on constructing Eulerian circuits from a weighted undirected graph is proposed. The principal stress lines are transformed into a weighted undirected graph, which divides the Eulerian subgraph by splitting the common nodes to plan the continuous fiber printing path circuit. Subsequently, the printing structure parameters are optimized based on simulation analysis. The experimental results, taking MBB (Messerschmitt-Bolkow-Blohm) beams as an example, show that the printing path designed in this paper improves the structural strength by 52.9% and the structural stiffness by 81.7% compared with the conventional printing path.

On the annihilator graphs of partial transformation semigroups

Let X n = { 1 , 2 , … , n } and P n be a partial transformation semigroup on X n . Obviously, the empty set ∅ is a zero element of P n and denoted by 0. Let P n * = P n \\ { 0 } . An element α ∈ P n is a zero divisor of P n if there exists β ∈ P n * such that α β = 0 = β α . The set Ann ( α ) = { β ∈ P n : α β = 0 = β α } is called the annihilator of α in P n . It is clear that 0 ∈ Ann ( α ) for all α ∈ P n . Let Z ( P n ) be the set of all zero divisors of P n and Z ( P n ) * = Z ( P n ) \\ { 0 } . Generally, if α ∈ Z ( P n ) , then there exists β ∈ Z ( P n ) * in which β ∈ Ann ( α ) . In this paper, we construct the annihilator graph Γ n of P n which is an undirected simple graph with vertex set Z ( P n ) * and two distinct vertices α and β are adjacent if and only if Ann ( α ) ∩ Ann ( β ) ≠ { 0 } . Furthermore, we prove some basic structural properties of Γ n and determine invariants for Γ n such as the diameter, girth, clique, domination number, and independence number.

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

The Parameterized Complexity of the Survivable Network Design Problem

For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph G with edge costs, a set R of terminal vertices, and an integer demand ds,t for every terminal pair s,t∈R. The task is to compute a subgraph H of G of minimum cost, such that there are at least ds,t disjoint paths between s and t in H. Depending on the type of disjointness we obtain several variants of SNDP that have been widely studied in the literature: if the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity variant (LC-SNDP), where the paths are disjoint on edges and non-terminals, i.e., they may only share terminals. In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size ℓ, the sum of demands D, the number of terminals k, and the maximum demand dmax.

Propagation of Hydraulic Fractures and Natural Fractures: The Bypassing Behavior in 3D Space

Summary The interaction between natural fractures (NFs) and hydraulic fractures (HFs) in 3D space poses significant challenges to numerical simulation. The interaction behaviors between HFs and NFs in 3D space include crossing, offset, stopping, and bypassing. Many existing numerical simulators are 2D, which inherently limits their ability to account for the vertical growth of fractures. Consequently, they are unable to model the bypassing behavior effectively. In this paper, a mathematical model for the propagation of HFs and NFs in a naturally fractured reservoir is established. The displacement discontinuity method (DDM) is used to solve the rock deformation, while the finite difference method (FDM) is utilized to solve the fluid flow within fractures. The undirected graph structure is used to represent the complex fracture network, and a dynamic adjustment of grid connectivity method is implemented to describe the process by which HFs bypass NFs. By integrating with graphics processing unit (GPU) computing, an efficient 3D simulator for HFs and NFs propagation is developed. The accuracy is verified against analytical solutions and reference solutions. Subsequently, a series of numerical studies on the bypassing behavior are conducted. The primary findings are as follows: (1) The simulator can accurately capture the interactions between HFs and NFs in 3D space, including crossing, arresting, and bypassing behaviors. (2) The bypassing behavior is characterized by three distinct processes—intersection, height growth of HF, and bypassing. (3) During the intersection stage, both the injection pressure and maximum NF width increase. During the height growth stage, the pressure is relatively stable, while the maximum NF width continues to increase. These two stages together result in the “storage” phenomenon. Once the bypassing behavior occurs, both the pressure and the maximum NF width decrease sharply, leading to a “release” phenomenon. Additionally, the “storage” and “release” phenomena may impact proppant transport. (4) Given the presence of bypassing behavior, it is essential to consider the NFs in 3D fracture simulators. The simulator and its findings can provide valuable insights for field design.

The First General Zagreb Index of the Zero Divisor Graph for the Ring Zpqk

This study investigates the application of graph theory in analyzing the zero divisor graph of a commutative ring, with a specific focus on its connection to the topological index. For an undirected graph Γ with consists of a non-empty set of vertices, V , and a set of edges, E, the first general Zagreb index is defined as a graph invariant that measures the sum of the degree of each vertex to the power of α= 0. Meanwhile, the zero divisor graph Γ of the commutative ring, R is the (undirected) graph with vertices the zero-divisors of R, and distinct vertices a and b are adjacent if and only if ab = 0. In this paper, the general formulas of the first general Zagreb index of the zero divisor graph for the ring of integers modulo pqk are computed for the cases δ = 1, 2, and 3. This research focuses on the ring defined as the integers modulo pqk, where k is a positive integer, p and q are primes p < q. Two examples are given to demonstrate the main f indings.

On Interdicting Dense Clusters in a Network

Given a vertex-weighted undirected graph with blocking costs of its vertices and edges, we seek a minimum cost subset of vertices and edges to block such that the weight of any γ-quasi-clique in the interdicted graph is at most some predefined threshold parameter. The value of [Formula: see text] specifies the edge density of cohesive vertex groups of interest in the network. The considered weighted γ-quasi-clique interdiction problem can be viewed as a natural generalization of several variations of the clique blocker problem previously studied in the literature. From the application perspective, this setting is primarily motivated by the problem of disrupting adversarial (“dark”) networks (e.g., social or communication networks), where γ-quasi-cliques represent “tightly knit” groups of adversaries that we aim to dismantle. We first address the theoretical computational complexity of the problem. We then exploit some basic characterization of its feasible solutions to derive a linear integer programming (IP) formulation. This linear IP model can be solved using a lazy-fashioned branch-and-cut scheme. We also propose a combinatorial branch-and-bound algorithm for solving this problem. The computational performance of the developed exact solution schemes is studied using a test bed of randomly generated and real-life networks. Finally, some interesting insights and observations are also provided using a well-known example of a terrorist network. History: Accepted by Russel Bent, Area Editor for Network Optimization: Algorithms & Applications. Funding: The work of S. Butenko was partially supported by the Air Force Office of Scientific Research under Award FA9550-23-1-0300. The work of O. A. Prokopyev was partially supported by the Office of Naval Research under Award ONR N00014-22-1-2678. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0027 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0027 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ . The online appendix is available at https://doi.org/10.1287/ijoc.2023.0027 .

Point-based Models: A Unified Approach for Geometric Constraint Analysis of Planar N-Bar Mechanisms with Rotary, Sliding, and Rolling Joints

Abstract Kinematic simulation of planar n-bar mechanisms has been an intense topic of study for several decades now. However, a large majority of efforts have focused on position analysis of such mechanisms with limited links and joint types. This paper presents a novel, unified approach to analysis of geometric constraints of planar n-bar mechanisms with revolute joint (R-joint), prismatic joint (P-joint), and rolling joint. This work is motivated by a need to create and program a system of constraint equations, which deal with different types of joints in a unified way. A key feature of this work is that the rolling joint constraints are represented by four-point models, which enables us to use the well-established undirected graph rigidity analysis algorithms. As a result, mechanisms with an arbitrary combination of revolute-, prismatic-joints, and wheel/gear/wheel-belt chains without any limitations on their actuation, can be analyzed and simulated efficiently for potential implementation in an interactive computer software and large-scale data generation.

Combining Genetic Algorithm with Local Search Method in Solving Optimization Problems

This research is focused on evolutionary algorithms, with genetic and memetic algorithms discussed in more detail. A graph theory problem related to finding a minimal Hamiltonian cycle in a complete undirected graph (Travelling Salesman Problem—TSP) is considered. The implementations of two approximate algorithms for solving this problem, genetic and memetic, are presented. The main objective of this study is to determine the influence of the local search method versus the influence of the genetic crossover operator on the quality of the solutions generated by the memetic algorithm for the same input data. The results show that when the number of possible Hamiltonian cycles in a graph is increased, the memetic algorithm finds better solutions. The execution time of both algorithms is comparable. Also, the number of solutions that mutated during the execution of the genetic algorithm exceeds 50% of the total number of all solutions generated by the crossover operator. In the memetic algorithm, the number of solutions that mutate does not exceed 10% of the total number of all solutions generated by the crossover operator, summed with those of the local search method.

Analysis of Graph Inverses and Algebraic Connectivity in a Systematic Way

an exhaustive and methodical study of algebraic connectedness and graph inverses. In many contexts, graph theory plays a crucial role, particularly when understanding intricate systems. This paper delves into the fundamental concepts of graph inverses and explains their significance for network analysis and connectivity evaluation. The Laplacian lattice's second-smallest eigenvalue, or algebraic connectivity µN−1, plays a crucial role in some features like network heartiness, synchronisation security, and diffusion processes. In this study, we focus on the algebraic connectedness in the network-of-networks (NoN), which is the general context of linked networks. A key component of science is the graph hypothesis. Algebraic graph hypothesis refers to the use of algebraic techniques to graph problems. This article concludes a focus on algebraic graph hypothesis and presents and examines a different algebraic and mathematical variety idea to regard as the greatest matching of an undirected graph.

POI VizNet: New QGIS Tool to construct Visibility Networks in Cities

Abstract. Network science and graph theory offer attractive models to describe urban phenomena and examine cases of urban planning and design. However, none of the existing network analysis tools consider simultaneously human visual perception and locations of urban activities. In this paper we introduce POI VizNet, a new QGIS plug-in that constructs various 2-dimentianal undirected graphs of unobstructed lines of sight. The plug-in integrates an increasing amount of available GIS-based data of Point Of Interest (POI) and visibility into one readily accessible analytical framework. Graphs are created between two types of decision points during urban travel – street intersections and POIs, origins, and destinations of travel, by connecting these potential observer’s decision locations in an open space between buildings. In addition, POI VizNet provides advanced options to build graphs using a predefined viewing distance and perceptual perspective. Visibility graphs are constructed and visualised as new layers in QGIS and delivered as network files suitable for further exploration, analysis, and visualisation in various network software packages.

Branch and Price for the Stochastic Traveling Salesman Problem with Generalized Latency

We consider an extension of the symmetric traveling salesman problem with generalized latency that explicitly models uncertainty. The stochastic traveling salesman problem with generalized latency (STSP-GL) aims to choose a subset of nodes of an undirected graph and determines a Hamiltonian tour among those nodes, minimizing an objective function that is a weighted combination of route design and passenger routing costs. These nodes are selected to ensure that a predefined percentage of uncertain passenger demand is served with a given probability. We formulate the STSP-GL as a stochastic program and propose a branch-and-price algorithm for solving its deterministic equivalent. We also develop a local search approach with which we improve the performance of the branch-and-price approach. We assess the efficiency of the proposed methods on a set of instances from the literature. We demonstrate that the proposed methods outperform a known benchmark, improving upper bounds by up to 85% and lower bounds by up to 55%. Finally, we show that solutions of the stochastic model are both more cost-effective and robust than those of the deterministic model. History: This paper has been accepted for the Transportation Science Special Issue on TSL Conference 2023. Funding: This work was supported by the Bundesministerium für Bildung und Forschung [Grant 03ZU1105FA]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0417 .

Path partitions of phylogenetic networks

In phylogenetics, evolution is traditionally represented in a tree-like manner. However, phylogenetic networks can be more appropriate for representing evolutionary events such as hybridization, horizontal gene transfer, and others. In particular, the class of forest-based networks was recently introduced to represent introgression, in which genes are swapped between species. A network is forest-based if it can be obtained by adding arcs to a collection of trees, so that the endpoints of the new arcs are in different trees. This contrasts with so-called tree-based networks, which are formed by adding arcs within a single tree.We are interested in the computational complexity of recognizing forest-based networks, which was recently left as an open problem by Huber et al. It has been observed that forest-based networks coincide with directed acyclic graphs that can be partitioned into induced paths, each ending at a leaf of the original graph. Several types of path partitions have been studied in the graph theory literature, but to our best knowledge this type of ‘leaf induced path partition’ has not been directly considered before. The study of forest-based networks in terms of these partitions allows us to establish closer relationships between phylogenetics and algorithmic graph theory, and to provide answers to problems in both fields.More specifically, we show that deciding whether a network is forest-based is NP-complete, even on input networks that are tree-based, binary, and have only three leaves. This shows that partitioning a directed acyclic graph into a constant number of induced paths is NP-complete, answering a recent question of Fernau et al. We then show that the problem is polynomial-time solvable on binary networks with two leaves and on the recently introduced class of orchards, which we show to be always forest-based. Finally, for undirected graphs, we introduce unrooted forest-based networks and provide hardness results for this class as well.