Undirected Graph Research Articles

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.

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.

On monoids of endomorphisms of a cycle graph

Abstract In this paper, we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids wEndCn and EndCn of all weak endomorphisms and all endomorphisms of an undirected cycle graph Cn with n vertices. We also describe Green’s relations and regularity of these monoids and calculate their cardinalities.

Distributed Bipartite Consensus of Multi-Agent Systems via Disturbance Rejection Control Strategy

This work aims to focus on analyzing the consensus control problem in cooperative–competitive networks in the occurrence of external disturbances. The primary motive of this work is to employ the equivalent input-disturbance estimation technique to compensate for the impact of external disturbances in the considered multi-agent system. In particular, a suitable low-pass filter is implemented to enhance the accuracy of disturbance estimation performance. In addition, a specific signed, connected, and structurally balanced undirected communication graph with positive and negative edge weights is considered to express the cooperation–competition communication among neighboring agents. The cooperative–competitive multi-agent system reaches its final state with same magnitude and in opposite direction under the considered structurally balanced graph. By utilizing the properties of Lyapunov stability theory and graph theory, the adequate conditions assuring the bipartite consensus of the examined multi-agent system are established as linear matrix inequalities. An illustrative example is delivered at the end to check the efficacy of the designed control scheme.

On Graphs of a Finite Full Transformation Semigroup

For n ≥ 2 , let T n denote the full transformation semigroup on X n = { 1 , … , n } , and let π denote the constant map defined by x π = 1 for all x ∈ X n . Consider the sets: L = { α ∈ T n : α β = π for some β ∈ T n ∖ { π } } , R = { α ∈ T n : γ α = π for some γ ∈ T n ∖ { π } } , T = L ∩ R and Z = L ∪ R . We define two undirected graphs, denoted by Γ ( T n ) and Ω ( T n ) , as simple graphs whose vertex sets are T ∖ { π } and Z ∖ { π } , and whose two distinct vertices α and β are adjacent if and only if α β = π or β α = π . We then investigate some properties of both Γ ( T n ) and Ω ( T n ) .

EDGE IRREGULAR REFLEXIVE LABELING OF DUMBBELL GRAPH, CORONA OF OPEN LADDER, AND NULL GRAPH

Graph is a simple, connected, undirected graph with vertex set and edge set . A graph is called to have an edge irregular reflexive -labeling if its vertices can be labeled with even numbers from until and its edges can be labeled with positive integers from to such that the weights for all the edges are different, where . The weight of edge uv in graph with labeling, denoted by , is defined as sum of the edge label and all vertex labels incident to that edge. The reflexive edge strength of a graph , denoted by , is the value of minimum of the largest label. In this paper, edge irregular reflexive -labeling for Dumbbell Graph and corona of open ladder and null graph will be determined. The reflexive edge strength of the Dumbbell Graph with and is for and for The reflexive edge strength of the corona of open ladder and null graph with n ≥ 3 and m ≥ 1 is for and for .

Triangle Centrality

Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. It has the distinct feature of allowing a vertex to be central if it is in many triangles or none at all. Given a simple, undirected graph \(G=(V,E)\) with \(n=|V|\) vertices and \(m=|E|\) edges, let \(\triangle(v)\) and \(\triangle(G)\) denote the respective triangle counts of \(v\) and \(G\) . Let \(N(v)\) be the neighborhood set of \(v\) . Respectively, \(N_{\triangle}(v)\) and \(N_{\triangle}[v]=\{v\}\cup N_{\triangle}(v)\) denote the set of neighbors that are in triangles with \(v\) and the closed set including \(v\) . Then the triangle centrality for a vertex \(v\) is \(\begin{align*}TC(v)=\frac{\frac{1}{3}\sum_{u\in N_{\triangle}[v]}\triangle(u)+\sum_{w\in\{N(v)\setminus N_{\triangle}(v)\}}\triangle(w)}{\triangle(G)}.\end{align*}\) We show experimentally that triangle centrality is broadly applicable to many different types of networks. Our empirical results demonstrate that 30% of the time triangle centrality identified central vertices that differed with those found by five well-known centrality measures, which suggests novelty without being overly specialized. It is also asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures. We introduce optimal algorithms that compute triangle centrality in \(O(m\overline{\delta})\) time and \(O(m+n)\) space, where \(\overline{\delta}\leq O(\sqrt{m})\) is the average degeneracy introduced by Burkhardt, Faber, and Harris (2020). In practical applications, \(\overline{\delta}\) is much smaller than \(\sqrt{m}\) so triangle centrality can be computed in nearly linear time. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Machine (PRAM), we give a near work-optimal parallel algorithm that takes \(O(\log n)\) time using \(O(m\sqrt{m})\) CREW PRAM processors. In MapReduce, we show it takes four rounds using \(O(m\sqrt{m})\) communication bits and is therefore optimal. We also derive a linear algebraic formulation of triangle centrality which can be computed in \(O(m\overline{\delta})\) time on sparse graphs.

THE REFLEXIVE EDGE STRENGTH OF THE PENTAGONAL SNAKE GRAPH AND CORONA OF THE OPEN TRIANGULAR LADDER AND NULL GRAPH

Assume that be an undirected simple graph with vertex set and edge set . The edge irregular reflexive -labeling of graph is a labeling selects positive integers from 1 to as edge labels and non negative even numbers from 0 to as vertex labels, and the weights assigned to each edge are distinct, where . On graph with labeling, the weight of edge is represented by which is defined as the sum of edge label and all vertex labels incident to that edge. Reflexive edge strength of graph is the minimum of the highest label, denoted by . In this research, reflexive edge strength for pentagonal snake graph and corona of open triangular ladder and null graph will be determined. The method of this research is literature study, the lower bound of determined by Ryan’s lemma and the upper bound by labeling. The reflexive edge strength of pentagonal snake graph with is for and for The reflexive edge strength of corona of open triangular ladder and null graph with n ≥ 3 and m ≥ 1 is and .

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored \(n\) -vertex undirected graph, vertices \(s\) and \(t\) , and an integer \(k\) , finds an \((s,t)\) -path containing at least \(k\) different colors in time \(2^{k}n^{\mathcal{O}(1)}\) . This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman on finding a path through \(k\) specified vertices. It also implies the first \(2^{k}n^{\mathcal{O}(1)}\) time algorithm for finding an \((s,t)\) -path of length at least \(k\) . Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an \(n\) -vertex undirected graph \(G\) , a matroid \(M\) whose elements correspond to the vertices of \(G\) and which is represented over a finite field of order \(q\) , a positive integer weight function on the vertices of \(G\) , two sets of vertices \(S,T\subseteq V(G)\) , and integers \(p,k,w\) , and the task is to find \(p\) vertex-disjoint paths from \(S\) to \(T\) so that the union of the vertices of these paths contains an independent set of \(M\) of cardinality \(k\) and weight \(w\) , while minimizing the sum of the lengths of the paths. We give a \(2^{p+\mathcal{O}(k^{2}\log(q+k))}n^{\mathcal{O}(1)}w\) time randomized algorithm for this problem.

Algorithms for the thief orienteering problem on directed acyclic graphs

We consider the scenario of routing an agent called a thief through a weighted graph G=(V,E) from a start vertex s to an end vertex t. A set I of items each with weight wi and profit pi is distributed among V∖{s,t}. The thief, who has a knapsack of capacity W, must follow a simple path from s to t within a given time T while packing in the knapsack a set of items taken from the vertices along the path of total weight at most W and maximum profit. The travel time across an edge depends on the edge length and current knapsack load.The thief orienteering problem (ThOP) is a generalization of the orienteering problem, the longest path problem, and the 0-1 knapsack problem. We prove that there exists no approximation algorithm for ThOP with constant approximation ratio unless P=NP, and we present a polynomial-time approximation scheme (PTAS) for a relaxed version of ThOP when G is directed and acyclic that produces solutions that use time at most T(1+ϵ) for any constant ϵ>0. We also present a fully polynomial-time approximation scheme (FPTAS) for ThOP on arbitrary undirected graphs where the travel time depends only on the lengths of the edges and T is the length of a shortest path from s to t plus a constant K. Finally, we present a FPTAS for a restricted version of the problem where the input graph is a clique.

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

On network deconvolution for undirected graphs.

Network deconvolution (ND) is a method to reconstruct a direct-effect network describing direct (or conditional) effects (or associations) between any two nodes from a given network depicting total (or marginal) effects (or associations). Its key idea is that, in a directed graph, a total effect can be decomposed into the sum of a direct and an indirect effects, with the latter further decomposed as the sum of various products of direct effects. This yields a simple closed-form solution for the direct-effect network, facilitating its important applications to distinguish direct and indirect effects. Despite its application to undirected graphs, it is not well known why the method works, leaving it with skepticism. We first clarify the implicit linear model assumption underlying ND, then derive a surprisingly simple result on the equivalence between ND and use of precision matrices, offering insightful justification and interpretation for the application of ND to undirected graphs. We also establish a formal result to characterize the effect of scaling a total-effect graph. Finally, leveraging large-scale genome-wide association study data, we show a novel application of ND to contrast marginal versus conditional genetic correlations between body height and risk of coronary artery disease; the results align with an inferred causal directed graph using ND. We conclude that ND is a promising approach with its easy and wide applicability to both directed and undirected graphs.

Redicolouring digraphs: Directed treewidth and cycle-degeneracy

Given a digraph D=(V,A) on n vertices and a vertex v∈V, the cycle-degree of v is the minimum size of a set S⊆V(D)∖{v} intersecting every directed cycle of D containing v. From this definition of cycle-degree, we define the c-degeneracy (or cycle-degeneracy) of D, which we denote by δc∗(D). It appears to be a nice generalisation of the undirected degeneracy. For instance, the dichromatic number χ→(D) of D is bounded above by δc∗(D)+1, where χ→(D) is the minimum integer k such that D admits a k-dicolouring, i.e., a partition of its vertices into k acyclic subdigraphs.In this work, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda’s conjecture (Cereceda, 2007) to digraphs. The k-dicolouring graph of D, denoted by Dk(D), is the undirected graph whose vertices are the k-dicolourings of D and in which two k-dicolourings are adjacent if they differ on the colour of exactly one vertex. This is a generalisation of the k-colouring graph of an undirected graph G, in which the vertices are the proper k-colourings of G.We show that Dk(D) has diameter at most Oδc∗(D)(nδc∗(D)+1) (respectively O(n2) and (δc∗(D)+1)n) when k is at least δc∗(D)+2 (respectively 32(δc∗(D)+1) and 2(δc∗(D)+1)). This improves known results on digraph redicolouring (Bousquet et al., 2023). Next, we extend a result due to Feghali (2021) to digraphs, showing that Dd+1(D) has diameter at most Od,ϵ(n(logn)d−1) when D has maximum average cycle-degree at most d−ϵ.We then show that two proofs of Bonamy and Bousquet (2018) for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph χ→g(D), which is the worst number of colours used in a greedy dicolouring. If k≥χ→g(D)+1, we show that Dk(D) has diameter at most 4⋅χ→(D)⋅n. The second one uses the D-width of a digraph, denoted by Dw(D), which is a generalisation of the treewidth to digraphs. If k≥Dw(D)+2, we show that Dk(D) has diameter at most 2(n2+n).Finally, we give a general theorem which makes a connection between the recolourability of a digraph D and the recolourability of its underlying graph UG(D). Assume that G is a class of undirected graphs, closed under edge-deletion and with bounded chromatic number, and let k≥χ(G) (i.e., k≥χ(G) for every G∈G) be such that, for every n-vertex graph G∈G, the diameter of the k-colouring graph of G is bounded by f(n) for some function f. We show that, for every n-vertex digraph D such that UG(D)∈G, the diameter of Dk(D) is bounded by 2f(n). For instance, this result directly extends a number of results on planar graph recolouring to planar digraph redicolouring.

The Use of Probability and Edge Analysis to Solve the Multi-Period Degree Constrained Minimum Spanning Tree Problem

The goal of the Multiperiod Degree Constrained Minimum Spanning Tree (MPDCMST) problem is to determine the smallest weight-spanning tree that satisfies the vertex installation criterion for each period and maintains the degree requirement in each vertex. This issue emerges as a network connection problem. The degree requirement indicates the reliability of each vertex, and the vertex connection/installation requirement denotes the priority vertices that must be inserted in the network within a specific time frame. The installation is split up into multiple phases/stages. This is because of various considerations such as severe weather, budgetary limitations, etc. In this research, two algorithms for solving the MPDCMST using probability hybridized with Prim’s modification, and edge analysis are proposed. The algorithms are implemented on the undirected complete graph of orders 10 to 100. The solutions are compared with some heuristics which are already in the literature. The results show that the proposed algorithms perform better.

Injective coloring of subclasses of chordal graphs

An injective k-coloring of a graph G=(V,E) is a function f:V→{1,2,…,k} such that for every pair of vertices u and v having a common neighbor, f(u)≠f(v). The injective chromatic number χi(G) of a graph G is the minimum k for which G admits an injective k-coloring. Given a graph G and a positive integer k, Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. Decide Injective Coloring Problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this result by proving that Decide Injective coloring Problem remains NP-complete for undirected path graphs, a proper subclass of chordal graphs. Moreover, we show that it is not possible to approximate the injective chromatic number of an undirected path graph within a factor of n1/3−ε in polynomial time for every ε>0 unless ZPP = NP. On the positive side, we prove that the injective chromatic number of an interval graph is either Δ(G) or Δ(G)+1, where Δ(G) is the maximum degree of G. We also characterize the interval graphs having χi(G)=Δ(G) and χi(G)=Δ(G)+1. As a consequence of this characterization, we obtain a linear time algorithm to find the injective chromatic number of an interval graph.

Single-exponential FPT algorithms for enumerating secluded [formula omitted]-free subgraphs and deleting to scattered graph classes

The celebrated notion of important separators bounds the number of small (S,T)-separators in a graph which is ‘farthest from S’ in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive, tailored to undirected graphs, that is phrased in terms of k-secluded vertex sets: sets with an open neighborhood of size at most k. In this terminology, the bound on important separators says that there are at most 4k maximal k-secluded connected vertex sets C containing S but disjoint from T. We generalize this statement significantly: even when we demand that G[C] avoids a finite set F of forbidden induced subgraphs, the number of such maximal subgraphs is 2O(k) and they can be enumerated efficiently. This enumeration algorithm allows us to give improved parameterized algorithms for Connectedk-SecludedF-Free Subgraph and for deleting into scattered graph classes.

ЩОДО ОДНОГО ПІДХОДУ ДО ВИРІШЕННЯ ПИТАННЯ АВТОМАТИЗОВАНОГО БУДУВАННЯ СХЕМИ ПОСИЛАНЬ РЕЛЯЦІЙНИХ БАЗ ДАНИХ

Introduction. The current trend of continuous integration of various areas of scientific and practical human activity contributes to the constant increase in the complexity of problems, the successful solution of which becomes possible only through the use of large volumes of information, as a rule, geographically distributed across multiple database files. In this case, the need to know the locations and features of the interaction of its components becomes of fundamental importance when organizing the most convenient and efficient ways to access it. In some cases, to meet such needs, you can use the services of existing products (Microsoft Access, Microsoft SQL Server, MySQL, FileMaker Pro, Oracle Database, dBASE, etc.). Sometimes, using a specialized SQL language or its dialects, it is more convenient to create a “handmade” one that will satisfy the stated requirements. If the user desires or needs to know the location of the information used in the work, understand the logic of the mechanism for obtaining it, and evaluate the effectiveness of the ways of its delivery to constantly monitor its current and future nomenclature and quantity, its developments can help, which can provide more detailed information, than known means. As experience shows, obtaining an integral and effective solution to this set of closely interrelated issues becomes possible only with the availability of a purely specialized software tool for the automated construction of link diagrams, which is a multifunctional, operational, easy-to-learn, and easy-to-use user environment. Endowed with the ability at any time and with minimal user participation to visualize on the screen the current structure of the created database with the required degree of clarity, detail, and ease of perception, such an environment should become a means of operational analysis and assessment of ways of possible modification of the database itself in the direction of creating a more effective one in practical terms, regarding the variant of its structure. Purpose. Research and practical testing of the existing capabilities of approaches, methods, algorithms, and software modules that were developed during the creation of a universal information software technology for working with undirected graphs of the “mixed forest” type. To evaluate the efficiency and feasibility of using these approaches to improve a set of integrated software tools for the automated construction of relational database link diagrams with minimal user participation in this process. Methods. Database systems, apparatus of graph-tree theory. Result. A new approach is proposed, and the result of its software implementation is presented in the form of a “pilot” version of a set of software tools for the automated construction of relational database link schemes. The basis of the approach is techniques, methods, algorithms, etc. which were created and implemented programmatically in a new, universal information software technology for working with (temporarily) undirected graphs of the “sour cream forest” type. , Software package (SP) makes “formal” the user’s participation in the process of building link diagrams. By greatly reducing the time required for this and providing more than 90% savings in the technical resources of the computer used, SP makes it possible to take into account data tables, the total number of fields of which is limited only by the latter’s capabilities. A positive conclusion about the suitability of such an approach will become an incentive for its further improvement in the form of specialized information software technology as an effective means of solving the problems of constructing relational database link schemes.