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Symmetry in Graph Algorithms and Graph Theory III

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Computer".

Deadline for manuscript submissions: 28 February 2025 | Viewed by 7179

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Special Issue Editor

Dr. Manuel Lafond

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Guest Editor

Department of Computer Science, Université de Sherbrooke, Sherbrooke, QC, Canada
Interests: algorithms; computational biology; graph theory; parameterized complexity; approximation algorithms
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Following the success of the second Special Issue of Symmetry entitled “Graph Algorithms and Graph Theory II”, it is my pleasure to be the Guest Editor for a third Special Issue.

Graphs have applications in numerous areas of computer science, including machine learning, computational biology, social network analysis, and many other areas, which all require fast algorithms for various optimization problems. Recent advancements in graph theory have shown that most graphs exhibit structural properties or symmetry that can be leveraged for the development of efficient algorithms. To cite a few examples, minor theory has paved the way for countless results in parameterized complexity, and several regularity lemmata have stimulated several new ideas in approximation algorithms. Moreover, these results only represent a fraction of the algorithmic applications of structural graph theory that have emerged over the last few decades. This demonstrates that expanding our fundamental knowledge of graphs, whether it be graphs in general or specific classes, is necessary in order to improve the state of the art in algorithms and complexity.

This Special Issue aims to improve our understanding of the interplay between algorithms, structure, and symmetry in graphs. The goal is to explore new directions in designing graph algorithms and to establish new foundations in structural graph theory.

The scope of the Special Issue includes, but is not limited to:

The design and analysis of graph algorithms, as well as parallel, randomized, parameterized, distributed, and other types of algorithms;
Structural graph theory with immediate or potential applications in algorithms and complexity analysis.

Dr. Manuel Lafond
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (6 papers)

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Research

27 pages, 1452 KiB

Open AccessArticle

Partial Imaginary Transition State (ITS) Graphs: A Formal Framework for Research and Analysis of Atom-to-Atom Maps of Unbalanced Chemical Reactions and Their Completions

by Marcos E. González Laffitte, Klaus Weinbauer, Tieu-Long Phan, Nora Beier, Nico Domschke, Christoph Flamm, Thomas Gatter, Daniel Merkle and Peter F. Stadler

Symmetry 2024, 16(9), 1217; https://doi.org/10.3390/sym16091217 - 16 Sep 2024

Viewed by 864

Abstract

Atom-to-atom maps (AAMs) are bijections that establish the correspondence of reactant and product atoms across chemical reactions. They capture crucial features of the reaction mechanism and thus play a central role in modeling chemistry at the level of graph transformations. AAMs are equivalent to so-called “imaginary transition state” (ITS) graphs, making it possible to reduce tasks such as the computational comparison of AAMs to testing graph isomorphisms. In many application scenarios, nonetheless, only partial information is available, i.e., only partial maps or, equivalently, only subgraphs of the ITS graphs, are known. Here, we investigate whether and how, and to what extent, such partial chemical data can be completed and compared. The focus of this contribution is entirely on the development of a solid mathematical foundation for the analysis of partial AAMs and their associated partial ITS graphs. Full article

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

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12 pages, 283 KiB

Open AccessArticle

Extending Ramsey Numbers for Connected Graphs of Size 3

by Emma Jent, Sawyer Osborn and Ping Zhang

Symmetry 2024, 16(8), 1092; https://doi.org/10.3390/sym16081092 - 22 Aug 2024

Viewed by 752

Abstract

It is well known that the famous Ramsey number

R (K_{3}, K_{3}) = 6

. That is, the minimum positive integer n for which every red-blue coloring of the edges of the complete graph

K_{n}

results in [...] Read more.

It is well known that the famous Ramsey number

R (K_{3}, K_{3}) = 6

. That is, the minimum positive integer n for which every red-blue coloring of the edges of the complete graph

K_{n}

results in a monochromatic triangle

K_{3}

is 6. It is also known that every red-blue coloring of

K_{6}

results in at least two monochromatic triangles, which need not be vertex-disjoint or edge-disjoint. This fact led to an extension of Ramsey numbers. For a graph F and a positive integer t, the vertex-disjoint Ramsey number

V R_{t} (F)

is the minimum positive integer n such that every red-blue coloring of the edges of the complete graph

K_{n}

of order n results in t pairwise vertex-disjoint monochromatic copies of subgraphs isomorphic to F, while the edge-disjoint Ramsey number

E R_{t} (F)

is the corresponding number for edge-disjoint subgraphs. Since

V R_{1} (F)

and

E R_{1} (F)

are the well-known Ramsey numbers of F, these new Ramsey concepts generalize the Ramsey numbers and provide a new perspective for this classical topic in graph theory. These numbers have been investigated for the two connected graphs

K_{3}

and the path

P_{3}

of order 3. Here, we study these numbers for the remaining connected graphs, namely, the path

P_{4}

and the star

K_{1, 3}

of size 3. We show that

V R_{t} (P_{4}) = 4 t + 1

for every positive integer t and

V R_{t} (K_{1, 3}) = 4 t

for every integer

t \geq 2

. For

t \leq 4

, the numbers

E R_{t} (K_{1, 3})

and

E R_{t} (P_{4})

are determined. These numbers provide information towards the goal of determining how the numbers

V R_{t} (F)

and

E R_{t} (F)

increase as t increases for each graph

F \in {K_{1, 3}, P_{4}}

. Full article

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

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29 pages, 3044 KiB

Open AccessArticle

Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory

by Agustín Moreno Cañadas, Ismael Gutierrez and Odette M. Mendez

Symmetry 2024, 16(5), 570; https://doi.org/10.3390/sym16050570 - 6 May 2024

Cited by 1 | Viewed by 1157

Abstract

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages. Full article

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

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15 pages, 1240 KiB

Open AccessArticle

Bipartite (P₆,C₆)-Free Graphs: Recognition and Optimization Problems

by Ruzayn Quaddoura and Ahmad Al-Qerem

Symmetry 2024, 16(4), 447; https://doi.org/10.3390/sym16040447 - 7 Apr 2024

Cited by 1 | Viewed by 834

Abstract

The canonical decomposition of a bipartite graph is a new decomposition method that involves three operators: parallel, series, and

K ⨁ S

. The class of weak-bisplit graphs is the class of totally decomposable graphs with respect to these operators, and the [...] Read more.

The canonical decomposition of a bipartite graph is a new decomposition method that involves three operators: parallel, series, and

K ⨁ S

. The class of weak-bisplit graphs is the class of totally decomposable graphs with respect to these operators, and the class of bicographs is the class of totally decomposable graphs with respect to parallel and series operators. We prove in this paper that the class of bipartite

(P_{6}, C_{6})

-free graphs is the class of bipartite graphs that are totally decomposable with respect to parallel and

K ⨁ S

operators. We present a linear time recognition algorithm for

(P_{6}, C_{6})

-free graphs that is symmetrical to the linear recognition algorithms of weak-bisplit graphs and

{s t a r}_{1,2, 3}

-free bipartite graphs. As a result of this algorithm, we present efficient solutions in this class of graphs for two optimization graph problems: the maximum balanced biclique problem and the maximum independent set problem. Full article

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

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17 pages, 6547 KiB

Open AccessArticle

Virtual Coordinate System Based on a Circulant Topology for Routing in Networks-On-Chip

by Andrei M. Sukhov, Aleksandr Y. Romanov and Maksim P. Selin

Symmetry 2024, 16(1), 127; https://doi.org/10.3390/sym16010127 - 21 Jan 2024

Cited by 2 | Viewed by 1303

Abstract

In this work, the circulant topology as an alternative to 2D mesh in networks-on-chip is considered. A virtual coordinate system for numbering nodes in the circulant topology is proposed, and the principle of greedy promotion is formulated. The rules for constructing the shortest routes between the two nodes based on coordinates are formulated. A technique for calculating optimal network configurations is described. Dense states of the network when all neighborhoods of the central node are filled with nodes and the network has the smallest diameter are defined. It is shown that with an equal number of nodes, the diameter of the circulant is two times smaller than the diameter of the 2D mesh. This is due to the large number of symmetries for the circulant, which leave the set of nodes unchanged. A comparison of communication stability in both topologies in the conditions of failure of network nodes is made, the network behavior under load and failures is modeled, and the advantages of the circulant topology are presented. Full article

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

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28 pages, 9032 KiB

Open AccessArticle

Modeling and Simulation of Physical Systems Formed by Bond Graphs and Multibond Graphs

by Gilberto Gonzalez-Avalos, Noe Barrera Gallegos, Gerardo Ayala-Jaimes, Aaron Padilla Garcia, Luis Flaviano Ferreyra García and Aldo Jesus Parente Rodríguez

Symmetry 2023, 15(12), 2170; https://doi.org/10.3390/sym15122170 - 6 Dec 2023

Viewed by 1320

Abstract

Current physical systems are built in more that one coordinate: for example, electrical power systems, aeronautical systems and robotic systems can be modeled in multibond graphs

(M B G)

. However, in these systems, some elements use only one axis or [...] Read more.

Current physical systems are built in more that one coordinate: for example, electrical power systems, aeronautical systems and robotic systems can be modeled in multibond graphs

(M B G)

. However, in these systems, some elements use only one axis or dimension—for example, actuators and controllers—which can be modeled in bond graphs

(B G)

. Therefore, in this paper, modeling of systems in multibond graphs and bond graphs (

M B G

B G

) is presented. Likewise, the junction structure of systems represented by (

M B G

B G

) is introduced. From this structure, mathematical modeling in the state space is presented. Likewise, modeling of systems on a platform (

M B G

B G

) can be seen as symmetric to the mathematical model that represents these systems. Finally, a synchronous generator modeled by (

M B G

B G

) as a case study is developed, and simulation results using 20-Sim software are shown. Furthermore, an electrical power system connected to the power supply of a DC motor as another case study is explained. Full article

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

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