Editorial

2 pages, 158 KiB

Open AccessEditorial

Graph Algorithms and Graph Theory-Symmetry Special Issue

by Manuel Lafond

Symmetry 2022, 14(8), 1748; https://doi.org/10.3390/sym14081748 - 22 Aug 2022

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Understanding the structure of graphs and the interplay between their parameters is a fundamental question that has been actively researched over the last century [...] Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

Research

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15 pages, 345 KiB

Open AccessArticle

Double Roman Domination in Generalized Petersen Graphs P(ck, k)

by Darja Rupnik Poklukar and Janez Žerovnik

Symmetry 2022, 14(6), 1121; https://doi.org/10.3390/sym14061121 - 29 May 2022

Cited by 5 | Viewed by 1338

Abstract

A double Roman dominating function on a graph

G = (V, E)

is a function

f : V \to {0, 1, 2, 3}

, satisfying the condition that every vertex u for which [...] Read more.

A double Roman dominating function on a graph

G = (V, E)

is a function

f : V \to {0, 1, 2, 3}

, satisfying the condition that every vertex u for which

f (u) = 1

is adjacent to at least one vertex assigned 2 or 3, and every vertex u with

f (u) = 0

is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2. The weight of f equals the sum

w (f) = \sum_{v \in V} f (v)

. The minimum weight of a double Roman dominating function of G is called the double Roman domination number

γ_{d R} (G)

of a graph G. We obtain tight bounds and in some cases closed expressions for the double Roman domination number of generalized Petersen graphs

P (c k, k)

. In short, we prove that

γ_{d R} (P (c k, k)) = \frac{3}{2} c k + ε

, where

lim_{c \to \infty, k \to \infty} \frac{ε}{c k} = 0

. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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15 pages, 1087 KiB

Open AccessArticle

Pancyclicity of the n-Generalized Prism over Skirted Graphs

by Artchariya Muaengwaeng, Ratinan Boonklurb and Sirirat Singhun

Symmetry 2022, 14(4), 816; https://doi.org/10.3390/sym14040816 - 14 Apr 2022

Cited by 1 | Viewed by 1170

Abstract

A side skirt is a planar rooted tree T,

T \neq P_{2}

, where the root of T is a vertex of degree at least two, and all other vertices except the leaves are of degree at least three. A reduced [...] Read more.

A side skirt is a planar rooted tree T,

T \neq P_{2}

, where the root of T is a vertex of degree at least two, and all other vertices except the leaves are of degree at least three. A reduced Halin graph or a skirted graph is a plane graph

G = T \cup P

, where T is a side skirt, and P is a path connecting the leaves of T in the order determined by the embedding of T. The structure of reduced Halin or skirted graphs contains both symmetry and asymmetry. For

n \geq 2

and

P_{n} = v_{1} v_{2} v_{3} \dots v_{n}

as a path of length

n - 1

, we call the Cartesian product of a graph G and a path

P_{n}

, the n-generalized prism over a graph G. We have known that the n-generalized prism over a skirted graph is Hamiltonian. To support the Bondy’s metaconjecture from 1971, we show that the n-generalized prism over a skirted graph is pancyclic. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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18 pages, 440 KiB

Open AccessArticle

Remarks on Parameterized Complexity of Variations of the Maximum-Clique Transversal Problem on Graphs

by Chuan-Min Lee

Symmetry 2022, 14(4), 676; https://doi.org/10.3390/sym14040676 - 24 Mar 2022

Cited by 4 | Viewed by 1274

Abstract

With the rapid growth in the penetration rate of mobile devices and the surge in demand for mobile data services, small cells and mobile backhaul networks have become the critical focus of next-generation mobile network development. Backhaul requirements within current wireless networks are [...] Read more.

With the rapid growth in the penetration rate of mobile devices and the surge in demand for mobile data services, small cells and mobile backhaul networks have become the critical focus of next-generation mobile network development. Backhaul requirements within current wireless networks are almost asymmetrical, with most traffic flowing from the core to the handset, but 5G networks will require more symmetrical backhaul capability. The deployment of small cells and the placement of transceivers for cellular phones are crucial in trading off the symmetric backhaul capability and cost-effectiveness. The deployment of small cells is related to the placement of transceivers for cellular phones. Chang, Kloks, and Lee transformed the placement problem into the maximum-clique transversal problem on graphs. From the theoretical point of view, our paper considers the parameterized complexity of variations of the maximum-clique transversal problem for split graphs, doubly chordal graphs, planar graphs, and graphs of bounded treewidth. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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15 pages, 1979 KiB

Open AccessArticle

Assembly Configuration Representation and Kinematic Modeling for Modular Reconfigurable Robots Based on Graph Theory

by Tuopu Zhang, Qinghao Du, Guilin Yang, Chongchong Wang, Chin-Yin Chen, Chi Zhang, Silu Chen and Zaojun Fang

Symmetry 2022, 14(3), 433; https://doi.org/10.3390/sym14030433 - 22 Feb 2022

Cited by 4 | Viewed by 1944

Abstract

A Modular Reconfigurable Robot (MRR) composed of standard joint and link modules has the advantages of rapid changeover of assembly configurations for a diversity of application requirements. To tackle the kinematics analysis issues for an MRR, it is essential to develop a generic [...] Read more.

A Modular Reconfigurable Robot (MRR) composed of standard joint and link modules has the advantages of rapid changeover of assembly configurations for a diversity of application requirements. To tackle the kinematics analysis issues for an MRR, it is essential to develop a generic method to mathematically represent all possible assembly configurations and automatically generate their kinematic models. In this paper, two types of robot modules, i.e., revolute joint modules and rigid link modules, are considered as the basic building blocks of an MRR. The topological structure of an MRR is represented by a graph, termed an Assembly Graph (AG), in which all robot modules, regardless of their types, are treated as vertices, while the connecting interfaces between any two robot modules are treated as edges. Based on graph theory, a modified adjacency matrix, termed an Assembly Adjacency Matrix (AAM), is proposed to represent the assembly configuration of an MRR, in which the three-dimensional assembly information is specified with connecting port vectors for the adjacent modules. A path matrix derived from the AAM of an MRR is employed to describe the connecting sequence of its constituting modules. The local frame representation of the Product-of-Exponentials (POE) formula is employed, which is configuration-independent. Therefore, for an MRR configuration represented with an AAM, its kinematic model will be automatically generated according to its path matrix. The proposed assembly configuration representation and kinematic modeling method are validated through a dual-branch MRR configuration. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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16 pages, 460 KiB

Open AccessArticle

Solving Robust Weighted Independent Set Problems on Trees and under Interval Uncertainty

by Ana Klobučar and Robert Manger

Symmetry 2021, 13(12), 2259; https://doi.org/10.3390/sym13122259 - 27 Nov 2021

Cited by 1 | Viewed by 1243

Abstract

The maximum weighted independent set (MWIS) problem is important since it occurs in various applications, such as facility location, selection of non-overlapping time slots, labeling of digital maps, etc. However, in real-life situations, input parameters within those models are often loosely defined or [...] Read more.

The maximum weighted independent set (MWIS) problem is important since it occurs in various applications, such as facility location, selection of non-overlapping time slots, labeling of digital maps, etc. However, in real-life situations, input parameters within those models are often loosely defined or subject to change. For such reasons, this paper studies robust variants of the MWIS problem. The study is restricted to cases where the involved graph is a tree. Uncertainty of vertex weights is represented by intervals. First, it is observed that the max–min variant of the problem can be solved in linear time. Next, as the most important original contribution, it is proved that the min–max regret variant is NP-hard. Finally, two mutually related approximation algorithms for the min–max regret variant are proposed. The first of them is already known, but adjusted to the considered situation, while the second one is completely new. Both algorithms are analyzed and evaluated experimentally. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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17 pages, 288 KiB

Open AccessArticle

All Graphs with a Failed Zero Forcing Number of Two

by Luis Gomez, Karla Rubi, Jorden Terrazas and Darren A. Narayan

Symmetry 2021, 13(11), 2221; https://doi.org/10.3390/sym13112221 - 20 Nov 2021

Cited by 5 | Viewed by 2031

Abstract

Given a graph G, the zero forcing number of G,

Z (G)

, is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. [...] Read more.

Given a graph G, the zero forcing number of G,

Z (G)

, is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graph and will be denoted by

F (G)

. We present new results involving this parameter. In particular, we completely characterize all graphs G where

F (G) = 2

, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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13 pages, 2476 KiB

Open AccessArticle

Online Activation and Deactivation of a Petri Net Supervisor

by Sadok Rezig, Nidhal Rezg and Zied Hajej

Symmetry 2021, 13(11), 2218; https://doi.org/10.3390/sym13112218 - 20 Nov 2021

Cited by 3 | Viewed by 1458

Abstract

This paper highlights algebraic and mathematical properties in symmetry with Petri nets in order to control automated systems such as flexible workshops, which represent one of the most important examples in industry and for discrete event systems in general. This project deals with [...] Read more.

This paper highlights algebraic and mathematical properties in symmetry with Petri nets in order to control automated systems such as flexible workshops, which represent one of the most important examples in industry and for discrete event systems in general. This project deals with the problem of forbidden state transition by using a new application of the theory of regions for supervisory control. In the literature, most control synthesis methods suffer greatly from a cumbersome calculation burden of the Petri net supervisor given the complex exploration of the state graph. Our new methodology lightens the computational load of the Petri net supervisor by choosing specific regions on the reachability graph, on which the control is calculated offline using CPLEX. The determined controller is activated online if the process enters the chosen region, and deactivated otherwise. All our experiments were applied in a flexible workshop implemented in our research laboratory, which was used to engrave selected models on glass blocks of different colors. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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8 pages, 311 KiB

Open AccessArticle

On the Johnson–Tzitzeica Theorem, Graph Theory, and Yang–Baxter Equations

by Florin F. Nichita

Symmetry 2021, 13(11), 2070; https://doi.org/10.3390/sym13112070 - 02 Nov 2021

Cited by 2 | Viewed by 1478

Abstract

This paper presents several types of Johnson–Tzitzeica theorems. Graph diagrams are used in this analysis. A symmetric scheme is derived, and new results are obtained and open problems stated. We also present results relating the graphs and the Yang–Baxter equation. This equation has [...] Read more.

This paper presents several types of Johnson–Tzitzeica theorems. Graph diagrams are used in this analysis. A symmetric scheme is derived, and new results are obtained and open problems stated. We also present results relating the graphs and the Yang–Baxter equation. This equation has certain symmetries, which are used in finding solutions for it. All these constructions are related to integrable systems. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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21 pages, 1385 KiB

Open AccessArticle

Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions

by Lissette Valdés, Alfonso Ariza, Sira M. Allende, Alicia Triviño and Gonzalo Joya

Symmetry 2021, 13(8), 1534; https://doi.org/10.3390/sym13081534 - 20 Aug 2021

Cited by 3 | Viewed by 1791

Abstract

A communication network management system takes the measurements of its state variables at specific instants of time, considering them constant in the interval between two consecutive measurements. Nevertheless, this assumption is not true, since these variables evolve in real time. Therefore, uncertainty is [...] Read more.

A communication network management system takes the measurements of its state variables at specific instants of time, considering them constant in the interval between two consecutive measurements. Nevertheless, this assumption is not true, since these variables evolve in real time. Therefore, uncertainty is inherent in the processing of the measurements during the intervals so that they cannot be efficiently managed using crisp variables. In this paper, we face this problem by modeling the communications network as a type-V fuzzy graph, where both the nodes and the links are described with precision, but the cost of each link is modeled as a triangular fuzzy number. Different fuzzy cost allocation functions and fuzzy optimization strategies are described and applied to the search for the shortest path between two nodes. An experimental study has been conducted using two representative networks: the backbone network of Nippon Telegraph and Telephone Corporation (NTT) and the National Science Foundation’s Network (NFSNET). In these networks, our fuzzy cost functions and strategies have been compared with the well-known crisp equivalents. The optimal search strategies are based on the proposed Fuzzy Dijkstra Algorithm (FDA), which is described deeply. The simulation results demonstrate that in all cases the fuzzy alternatives surpass or equal the crisp equivalents with statistically significant values. Specifically, the so-called Strategy 8 presents the best throughput, as it significantly exceeds the performance of all those evaluated, achieving a Global Mean Delivery Rate (GMDR) close to 1. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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10 pages, 347 KiB

Open AccessArticle

Connectivity of Semiring Valued Graphs

by Shyam Sundar Santra, Prabhakaran Victor, Mahadevan Chandramouleeswaran, Rami Ahmad El-Nabulsi, Khaled Mohamed Khedher and Vediyappan Govindan

Symmetry 2021, 13(7), 1227; https://doi.org/10.3390/sym13071227 - 08 Jul 2021

Cited by 1 | Viewed by 1684

Abstract

Graph connectivity theory is important in network implementations, transportation, network routing and network tolerance, among other things. Separation edges and vertices refer to single points of failure in a network, and so they are often sought-after. Chandramouleeswaran et al. introduced the principle of [...] Read more.

Graph connectivity theory is important in network implementations, transportation, network routing and network tolerance, among other things. Separation edges and vertices refer to single points of failure in a network, and so they are often sought-after. Chandramouleeswaran et al. introduced the principle of semiring valued graphs, also known as S-valued symmetry graphs, in 2015. Since then, works on S-valued symmetry graphs such as vertex dominating set, edge dominating set, regularity, etc. have been done. However, the connectivity of S-valued graphs has not been studied. Motivated by this, in this paper, the concept of connectivity in S-valued graphs has been studied. We have introduced the term vertex S-connectivity and edge S-connectivity and arrived some results for connectivity of a complete S-valued symmetry graph, S-path and S-star. Unlike the graph theory, we have observed that the inequality for connectivity

κ (G) \leq κ^{'} (G) \leq δ (G)

holds in the case of S-valued graphs only when there is a symmetry of the graph as seen in Examples 3–5. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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12 pages, 1744 KiB

Open AccessArticle

Automatic Unsupervised Texture Recognition Framework Using Anisotropic Diffusion-Based Multi-Scale Analysis and Weight-Connected Graph Clustering

by Tudor Barbu

Symmetry 2021, 13(6), 925; https://doi.org/10.3390/sym13060925 - 23 May 2021

Cited by 4 | Viewed by 1865

Abstract

A novel unsupervised texture classification technique is proposed in this research work. The proposed method clusters automatically the textures of an image collection in similarity classes whose number is not a priori known. A nonlinear diffusion-based multi-scale texture analysis approach is introduced first. [...] Read more.

A novel unsupervised texture classification technique is proposed in this research work. The proposed method clusters automatically the textures of an image collection in similarity classes whose number is not a priori known. A nonlinear diffusion-based multi-scale texture analysis approach is introduced first. It creates an effective scale-space by using a well-posed anisotropic diffusion filtering model that is proposed and approximated numerically here. A feature extraction process using a bank of circularly symmetric 2D filters is applied at each scale, then a rotation-invariant texture feature vector is achieved for the current image by combining the feature vectors computed at all these scales. Next, a weighted similarity graph, whose vertices correspond to the texture feature vectors and the weights of its edges are obtained from the distances computed between these vectors, is created. A novel weighted graph clustering technique is then applied to this similarity graph, to determine the texture classes. Numerical simulations and method comparisons illustrating the effectiveness of the described framework are also discussed in this work. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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