Editorial

2 pages, 144 KiB

Open AccessEditorial

by Jose M. Rodriguez

Symmetry 2018, 10(1), 32; https://doi.org/10.3390/sym10010032 - 22 Jan 2018

Cited by 2 | Viewed by 3416

This book contains the successful invited submissions [1–10] to a special issue of Symmetry on the subject area of ‘graph theory’ [...]
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(This article belongs to the Special Issue Graph Theory)

Research

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20 pages, 905 KiB

Open AccessArticle

Efficient Location of Resources in Cylindrical Networks

by José Juan Carreño, José Antonio Martínez and María Luz Puertas

Symmetry 2018, 10(1), 24; https://doi.org/10.3390/sym10010024 - 10 Jan 2018

Cited by 4 | Viewed by 3230

Abstract

The location of resources in a network satisfying some optimization property is a classical combinatorial problem that can be modeled and solved by using graphs. Key tools in this problem are the domination-type properties, which have been defined and widely studied in different [...] Read more.

The location of resources in a network satisfying some optimization property is a classical combinatorial problem that can be modeled and solved by using graphs. Key tools in this problem are the domination-type properties, which have been defined and widely studied in different types of graph models, such as undirected and directed graphs, finite and infinite graphs, simple graphs and hypergraphs. When the required optimization property is that every node of the network must have access to exactly one node with the desired resource, the appropriate models are the efficient dominating sets. However, the existence of these vertex sets is not guaranteed in every graph, so relaxing some conditions is necessary to ensure the existence of some kind of dominating sets, as efficient as possible, in a larger number of graphs. In this paper, we study independent

[1, 2]

-sets, a generalization of efficient dominating sets defined by Chellali et al., in the case of cylindrical networks. It is known that efficient dominating sets exist in very special cases of cylinders, but the particular symmetry of these graphs will allow us to provide regular patterns that guarantee the existence of independent

[1, 2]

-sets in every cylinder, except in one single case, and to compute exact values of the optimal parameter, the independent

[1, 2]

-number, in cylinders of selected sizes. Full article

(This article belongs to the Special Issue Graph Theory)

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4600 KiB

Open AccessArticle

Analyzing Spatial Behavior of Backcountry Skiers in Mountain Protected Areas Combining GPS Tracking and Graph Theory

by Karolina Taczanowska, Mikołaj Bielański, Luis-Millán González, Xavier Garcia-Massó and José L. Toca-Herrera

Symmetry 2017, 9(12), 317; https://doi.org/10.3390/sym9120317 - 14 Dec 2017

Cited by 22 | Viewed by 5888

Abstract

Mountain protected areas (PAs) aim to preserve vulnerable environments and at the same time encourage numerous outdoor leisure activities. Understanding the way people use natural environments is crucial to balance the needs of visitors and site capacities. This study aims to develop an [...] Read more.

Mountain protected areas (PAs) aim to preserve vulnerable environments and at the same time encourage numerous outdoor leisure activities. Understanding the way people use natural environments is crucial to balance the needs of visitors and site capacities. This study aims to develop an approach to evaluate the structure and use of designated skiing zones in PAs combining Global Positioning System (GPS) tracking and analytical methods based on graph theory. The study is based on empirical data (n = 609 GPS tracks of backcountry skiers) collected in Tatra National Park (TNP), Poland. The physical structure of the entire skiing zones system has been simplified into a graph structure (structural network; undirected graph). In a second step, the actual use of the area by skiers (functional network; directed graph) was analyzed using a graph-theoretic approach. Network coherence (connectivity indices: β, γ, α), movement directions at path segments, and relative importance of network nodes (node centrality measures: degree, betweenness, closeness, and proximity prestige) were calculated. The system of designated backcountry skiing zones was not evenly used by the visitors. Therefore, the calculated parameters differ significantly between the structural and the functional network. In particular, measures related to the actually used trails are of high importance from the management point of view. Information about the most important node locations can be used for planning sign-posts, on-site maps, interpretative boards, or other tourist infrastructure. Full article

(This article belongs to the Special Issue Graph Theory)

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3495 KiB

Open AccessArticle

Reconstructing Damaged Complex Networks Based on Neural Networks

by Ye Hoon Lee and Insoo Sohn

Symmetry 2017, 9(12), 310; https://doi.org/10.3390/sym9120310 - 09 Dec 2017

Cited by 9 | Viewed by 3626

Abstract

Despite recent progress in the study of complex systems, reconstruction of damaged networks due to random and targeted attack has not been addressed before. In this paper, we formulate the network reconstruction problem as an identification of network structure based on much reduced [...] Read more.

Despite recent progress in the study of complex systems, reconstruction of damaged networks due to random and targeted attack has not been addressed before. In this paper, we formulate the network reconstruction problem as an identification of network structure based on much reduced link information. Furthermore, a novel method based on multilayer perceptron neural network is proposed as a solution to the problem of network reconstruction. Based on simulation results, it was demonstrated that the proposed scheme achieves very high reconstruction accuracy in small-world network model and a robust performance in scale-free network model. Full article

(This article belongs to the Special Issue Graph Theory)

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3339 KiB

Open AccessArticle

Graphical Classification in Multi-Centrality-Index Diagrams for Complex Chemical Networks

by Yasutaka Mizui, Tetsuya Kojima, Shigeyuki Miyagi and Osamu Sakai

Symmetry 2017, 9(12), 309; https://doi.org/10.3390/sym9120309 - 09 Dec 2017

Cited by 18 | Viewed by 4713

Abstract

Various sizes of chemical reaction network exist, from small graphs of linear networks with several inorganic species to huge complex networks composed of protein reactions or metabolic systems. Huge complex networks of organic substrates have been well studied using statistical properties such as [...] Read more.

Various sizes of chemical reaction network exist, from small graphs of linear networks with several inorganic species to huge complex networks composed of protein reactions or metabolic systems. Huge complex networks of organic substrates have been well studied using statistical properties such as degree distributions. However, when the size is relatively small, statistical data suffers from significant errors coming from irregular effects by species, and a macroscopic analysis is frequently unsuccessful. In this study, we demonstrate a graphical classification method for chemical networks that contain tens of species. Betweenness and closeness centrality indices of a graph can create a two-dimensional diagram with information of node distribution for a complex chemical network. This diagram successfully reveals systematic sharing of roles among species as a semi-statistical property in chemical reactions, and distinguishes it from the ones in random networks, which has no functional node distributions. This analytical approach is applicable for rapid and approximate understanding of complex chemical network systems such as plasma-enhanced reactions as well as visualization and classification of other graphs. Full article

(This article belongs to the Special Issue Graph Theory)

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3286 KiB

Open AccessArticle

Analytical Treatment of Higher-Order Graphs: A Path Ordinal Method for Solving Graphs

by Hala Kamal, Alicia Larena and Eusebio Bernabeu

Symmetry 2017, 9(11), 288; https://doi.org/10.3390/sym9110288 - 22 Nov 2017

Cited by 1 | Viewed by 3166

Abstract

Analytical treatment of the composition of higher-order graphs representing linear relations between variables is developed. A path formalism to deal with problems in graph theory is introduced. It is shown how paths in the composed graph representing individual contributions to variables relation can [...] Read more.

Analytical treatment of the composition of higher-order graphs representing linear relations between variables is developed. A path formalism to deal with problems in graph theory is introduced. It is shown how paths in the composed graph representing individual contributions to variables relation can be enumerated and represented by ordinals. The method allows for one to extract partial information and gives an alternative to classical graph approach. Full article

(This article belongs to the Special Issue Graph Theory)

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776 KiB

Open AccessArticle

Mathematical Properties on the Hyperbolicity of Interval Graphs

by Juan C. Hernández-Gómez, Rosalío Reyes, José M. Rodríguez and José M. Sigarreta

Symmetry 2017, 9(11), 255; https://doi.org/10.3390/sym9110255 - 01 Nov 2017

Cited by 5 | Viewed by 3021

Abstract

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs [...] Read more.

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1. Full article

(This article belongs to the Special Issue Graph Theory)

818 KiB

Open AccessArticle

β-Differential of a Graph

by Ludwin A. Basilio, Sergio Bermudo, Jesús Leaños and José M. Sigarreta

Symmetry 2017, 9(10), 205; https://doi.org/10.3390/sym9100205 - 30 Sep 2017

Cited by 7 | Viewed by 3220

Abstract

Let

G = (V, E)

be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let

B (D)

be the set of neighbours of D in [...] Read more.

Let

G = (V, E)

be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let

B (D)

be the set of neighbours of D in

V ∖ D

. The differential

\partial (D)

of D is defined as

| B (D) | - | D |

. The maximum value of

\partial (D)

taken over all subsets

D \subseteq V

is the differential

\partial (G)

of G. For

β \in (- 1, Δ)

, the β-differential

\partial_{β} (G)

of G is the maximum value of

{| B (D) | - β | D | : D \subseteq V}

. Motivated by an influential maximization problem, in this paper we study the

β

-differential of G. Full article

(This article belongs to the Special Issue Graph Theory)

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562 KiB

Open AccessArticle

Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

by Álvaro Martínez-Pérez

Symmetry 2017, 9(10), 199; https://doi.org/10.3390/sym9100199 - 24 Sep 2017

Cited by 7 | Viewed by 3553

Abstract

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is

δ

-hyperbolic if every geodesic triangle is

δ

-thin. In this paper, [...] Read more.

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is

δ

-hyperbolic if every geodesic triangle is

δ

-thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons. Full article

(This article belongs to the Special Issue Graph Theory)

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336 KiB

Open AccessFeature PaperArticle

The Simultaneous Local Metric Dimension of Graph Families

by Gabriel A. Barragán-Ramírez, Alejandro Estrada-Moreno, Yunior Ramírez-Cruz and Juan A. Rodríguez-Velázquez

Symmetry 2017, 9(8), 132; https://doi.org/10.3390/sym9080132 - 27 Jul 2017

Cited by 4 | Viewed by 3330

Abstract

In a graph

G = (V, E)

, a vertex

v \in V

is said to distinguish two vertices x and y if

d_{G} (v, x) \neq d_{G} (v, y)

. A [...] Read more.

In a graph

G = (V, E)

, a vertex

v \in V

is said to distinguish two vertices x and y if

d_{G} (v, x) \neq d_{G} (v, y)

. A set

S \subseteq V

is said to be a local metric generator for G if any pair of adjacent vertices of G is distinguished by some element of S. A minimum local metric generator is called a local metric basis and its cardinality the local metric dimension of G. A set

S \subseteq V

is said to be a simultaneous local metric generator for a graph family

G = {G_{1}, G_{2}, \dots, G_{k}}

, defined on a common vertex set, if it is a local metric generator for every graph of the family. A minimum simultaneous local metric generator is called a simultaneous local metric basis and its cardinality the simultaneous local metric dimension of

G

. We study the properties of simultaneous local metric generators and bases, obtain closed formulae or tight bounds for the simultaneous local metric dimension of several graph families and analyze the complexity of computing this parameter. Full article

(This article belongs to the Special Issue Graph Theory)

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850 KiB

Open AccessFeature PaperArticle

Gromov Hyperbolicity in Mycielskian Graphs

by Ana Granados, Domingo Pestana, Ana Portilla and José M. Rodríguez

Symmetry 2017, 9(8), 131; https://doi.org/10.3390/sym9080131 - 27 Jul 2017

Cited by 5 | Viewed by 3808

Abstract

Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph

G^{M}

is hyperbolic and that [...] Read more.

Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph

G^{M}

is hyperbolic and that

δ (G^{M})

is comparable to

diam (G^{M})

. Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs; in fact, it is shown that

5 / 4 \leq δ (G^{M}) \leq 5 / 2

. Graphs G whose Mycielskian have hyperbolicity constant

5 / 4

or

5 / 2

are characterized. The hyperbolicity constants of the Mycielskian of path, cycle, complete and complete bipartite graphs are calculated explicitly. Finally, information on

δ (G)

just in terms of

δ (G^{M})

is obtained. Full article

(This article belongs to the Special Issue Graph Theory)

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