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

Deadline for manuscript submissions: closed (31 January 2022) | Viewed by 9509

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Special Issue Editor

Dr. Magdalena Lemańska

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Guest Editor

Technical Physics and Applied Mathematics Department, Gdańsk University of Technology, 80-803 Gdańsk, Poland
Interests: domination topics of graph theory

Special Issue Information

Dear Colleagues,

Symmetry is one of the most important (not only aesthetic) criteria that ilustrate the structure and properties of graphs. Very often in graph theory, graphs are drawn symmetrically. There are various criteria for describing a graph as "symmetric" and describing such symmetric graphs has been the subject of much research. In recent years, the role of symmetry in graph theory significantly increased and has also covered such areas as metric dimension of graphs, domination theory, graph colourings resolving sets or independent sets in graphs. Potential topics of the Special Issue include but are not limited to the above areas, both from a theoretical and an applied point of view.

Dr. Magdalena Lemańska
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.

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Keywords

metric dimension of graphs
domination theory
graph colourings
resolving sets in graphs
symmetric graphs
independent sets

Published Papers (6 papers)

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Research

15 pages, 2490 KiB

Open AccessArticle

Extended Graph of the Fuzzy Topographic Topological Mapping Model

by Muhammad Zillullah Mukaram, Tahir Ahmad, Norma Alias, Noorsufia Abd Shukor and Faridah Mustapha

Symmetry 2021, 13(11), 2203; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112203 - 18 Nov 2021

Cited by 2 | Viewed by 1118

Abstract

Fuzzy topological topographic mapping (

F T T M

) is a mathematical model which consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM,

F T T M_{n}

, is [...] Read more.

Fuzzy topological topographic mapping (

F T T M

) is a mathematical model which consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM,

F T T M_{n}

, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of

F T T M

, namely the homeomorphisms between its components, allows the generation of new

F T T M

. The generated

F T T M

s can be represented as pseudo graphs. A graph of pseudo degree zero is a special type of graph where each of the

F T T M

components differs from the one adjacent to it. Previous researchers have investigated and conjectured the number of generated

F T T M

pseudo degree zero with respect to n number of components and k number of versions. In this paper, the conjecture is proven analytically for the first time using a newly developed grid-based method. Some definitions and properties of the novel grid-based method are introduced and developed along the way. The developed definitions and properties of the method are then assembled to prove the conjecture. The grid-based technique is simple yet offers some visualization features of the conjecture. Full article

(This article belongs to the Special Issue Research on Symmetry Applied in Graph Theory)

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13 pages, 290 KiB

Open AccessArticle

On the Total Neighbor Sum Distinguishing Index of IC-Planar Graphs

by Donghan Zhang, Chao Li and Fugang Chao

Symmetry 2021, 13(10), 1787; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101787 - 26 Sep 2021

Cited by 1 | Viewed by 1049

Abstract

A proper total k-coloring

ϕ

of G with

\sum_{z \in E_{G} (u) \cup {u}} ϕ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} ϕ (z)

for [...] Read more.

A proper total k-coloring

ϕ

of G with

\sum_{z \in E_{G} (u) \cup {u}} ϕ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} ϕ (z)

for each

u v \in E (G)

is called a total neighbor sum distinguishing k-coloring, where

E_{G} (u) = {u v | u v \in E (G)}

. Pilśniak and Woźniak conjectured that every graph with maximum degree

Δ

exists a total neighbor sum distinguishing

(Δ + 3)

-coloring. In this paper, we proved that any IC-planar graph with

Δ \geq 12

satisfies this conjecture, which improves the result of Song and Xu. Full article

(This article belongs to the Special Issue Research on Symmetry Applied in Graph Theory)

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12 pages, 382 KiB

Open AccessArticle

Coloring Properties of Mixed Cycloids

by György Dósa, Nicholas Newman, Zsolt Tuza and Vitaly Voloshin

Symmetry 2021, 13(8), 1539; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081539 - 21 Aug 2021

Cited by 2 | Viewed by 1749

Abstract

In this paper, we investigate partitions of highly symmetrical discrete structures called cycloids. In general, a mixed hypergraph has two types of hyperedges. The vertices are colored in such a way that each C-edge has two vertices of the same color, and each D-edge has two vertices of distinct colors. In our case, a mixed cycloid is a mixed hypergraph whose vertices can be arranged in a cyclic order, and every consecutive p vertices form a C-edge, and every consecutive q vertices form a D-edge in the ordering. We completely determine the maximum number of colors that can be used for any

p \geq 3

and any

q \geq 2

. We also develop an algorithm that generates a coloring with any number of colors between the minimum and maximum. Finally, we discuss the colorings of mixed cycloids when the maximum number of colors coincides with its upper bound, which is the largest cardinality of a set of vertices containing no C-edge. Full article

(This article belongs to the Special Issue Research on Symmetry Applied in Graph Theory)

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10 pages, 342 KiB

Open AccessArticle

Common Independence in Graphs

by Magda Dettlaff, Magdalena Lemańska and Jerzy Topp

Symmetry 2021, 13(8), 1411; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081411 - 02 Aug 2021

Cited by 3 | Viewed by 1536

Abstract

The cardinality of a largest independent set of G, denoted by

α (G)

, is called the independence number of G. The independent domination number

i (G)

of a graph G is the cardinality of a smallest [...] Read more.

The cardinality of a largest independent set of G, denoted by

α (G)

, is called the independence number of G. The independent domination number

i (G)

of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by

α_{c} (G)

, as the greatest integer r such that every vertex of G belongs to some independent subset X of

V_{G}

with

| X | \geq r

. The common independence number

α_{c} (G)

of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality

α_{c} (G)

in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities

i (G) \leq α_{c} (G) \leq α (G)

. In this paper, we characterize the trees T for which

i (T) = α_{c} (T)

, and the block graphs G for which

α_{c} (G) = α (G)

. Full article

(This article belongs to the Special Issue Research on Symmetry Applied in Graph Theory)

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9 pages, 904 KiB

Open AccessFeature PaperArticle

Facial Homogeneous Colouring of Graphs

by Tomáš Madaras and Mária Šurimová

Symmetry 2021, 13(7), 1213; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071213 - 06 Jul 2021

Viewed by 1223

Abstract

A proper colouring of a plane graph G is called facially homogeneous if it uses the same number of colours for every face of G. We study various sufficient conditions of facial homogeneous colourability of plane graphs, its relation to other facial [...] Read more.

(This article belongs to the Special Issue Research on Symmetry Applied in Graph Theory)

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12 pages, 331 KiB

Open AccessArticle

On 2-Rainbow Domination Number of Generalized Petersen Graphs P(5k,k)

by Rija Erveš and Janez Žerovnik

Symmetry 2021, 13(5), 809; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050809 - 06 May 2021

Cited by 4 | Viewed by 1585

Abstract

We obtain new results on 2-rainbow domination number of generalized Petersen graphs

P (5 k, k)

. In some cases (for some infinite families), exact values are established, and in all other cases lower and upper bounds are given. In [...] Read more.

We obtain new results on 2-rainbow domination number of generalized Petersen graphs

P (5 k, k)

. In some cases (for some infinite families), exact values are established, and in all other cases lower and upper bounds are given. In particular, it is shown that, for

k > 3

γ_{r 2} (P (5 k, k)) = 4 k

for

k \equiv 2, 8 mod 10

γ_{r 2} (P (5 k, k)) = 4 k + 1

for

k \equiv 5, 9 mod 10

4 k + 1 \leq γ_{r 2} (P (5 k, k)) \leq 4 k + 2

for

k \equiv 1, 6, 7 mod 10

, and

4 k + 1 \leq γ_{r 2} (P (5 k, k)) \leq 4 k + 3

for

k \equiv 0, 3, 4 mod 10

. Full article

(This article belongs to the Special Issue Research on Symmetry Applied in Graph Theory)

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