Search results for: neighborhood corona graph

Authors: U. Mary

Abstract:

Let be a graph with a finite, nonempty set of objects called vertices together with a set of unordered pairs of distinct vertices of called edges. The vertex set is denoted by and the edge set by. Given two graphs and the wiener index of, wiener index for the splitting graph of a graph, the first Zagreb index of and its splitting graph, the 3-steiner wiener index of, the 3-steiner wiener index of a special graph are explored in this paper.

Keywords: complementary prism graph, first Zagreb index, neighborhood corona graph, steiner distance, splitting graph, steiner wiener index, wiener index

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Authors: A. Kulandai Therese

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The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics.In this paper, we present the explicit eccentric connectivity index, first and second Zagreb indices for a Corona graph and sub division-related corona graphs.

Keywords: corona graph, degree, eccentricity, eccentric connectivity index, first zagreb index, second zagreb index, subdivision graphs

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Authors: Gowtham Kalkere Jayanna, Mohamad Nazri Husin

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Graph theory, chemistry, and technology are all combined in cheminformatics. The structure and physiochemical properties of organic substances are linked using some useful graph invariants and the corresponding molecular graph. In this paper, we study specific reverse topological indices such as the reverse sum-connectivity index, the reverse Zagreb index, the reverse arithmetic-geometric, and the geometric-arithmetic, the reverse Sombor, the reverse Nirmala indices for the bistar graphs B (n: m) and the corona product Kₘ∘Kₙ', where Kₙ' Represent the complement of a complete graph Kₙ.

Keywords: reverse topological indices, bistar graph, the corona product, graph

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Authors: Soner Nandappa D., Ahmed Mohammed Naji

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In a graph G = (V, E), the distance from a vertex v to a vertex u is the length of shortest v to u path. The eccentricity e(v) of v is the distance to a farthest vertex from v. The diameter diam(G) is the maximum eccentricity. The k-distance neighborhood of v, for 0 ≤ k ≤ e(v), is Nk(v) = {u ϵ V (G) : d(v, u) = k}. In this paper, we introduce a new distance degree based topological polynomial of a graph G is called a k- distance neighborhood polynomial, denoted Nk(G, x). It is a polynomial with the coefficient of the term k, for 0 ≤ k ≤ e(v), is the sum of the cardinalities of Nk(v) for every v ϵ V (G). Some properties of k- distance neighborhood polynomials are obtained. Exact formulas of the k- distance neighborhood polynomial for some well-known graphs, Cartesian product and join of graphs are presented.

Keywords: vertex degrees, distance in graphs, graph operation, Nk-polynomials

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Authors: Vadim E. Levit, Eugen Mandrescu

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A graph is well-covered if all its maximal independent sets are of the same size. A well-covered graph is 1-well-covered if deletion of every vertex of the graph leaves it well-covered. It is known that a graph without isolated vertices is 1-well-covered if and only if every two disjoint independent sets are included in two disjoint maximum independent sets. Well-covered graphs are related to combinatorial commutative algebra (e.g., every Cohen-Macaulay graph is well-covered, while each Gorenstein graph without isolated vertices is 1-well-covered). Our intent is to construct several infinite families of 1-well-covered graphs using the following known graph operations: corona, join, and rooted product of graphs. Adopting some known techniques used to advantage for well-covered graphs, one can prove that: if the graph G has no isolated vertices, then the corona of G and H is 1-well-covered if and only if H is a complete graph of order two at least; the join of the graphs G and H is 1-well-covered if and only if G and H have the same independence number and both are 1-well-covered; if H satisfies the property that every three pairwise disjoint independent sets are included in three pairwise disjoint maximum independent sets, then the rooted product of G and H is 1-well-covered, for every graph G. These findings show not only how to generate some more families of 1-well-covered graphs, but also that, to this aim, sometimes, one may use graphs that are not necessarily 1-well-covered.

Keywords: maximum independent set, corona, concatenation, join, well-covered graph

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Authors: David Pham, Yongfeng Zhang

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Graphs are an important structure for data storage and computation. Recent years have seen the success of deep learning on graphs such as Graph Neural Networks (GNN) on various data mining and machine learning tasks. However, most of the deep learning models on graphs cannot easily explain their predictions and are thus often labelled as “black boxes.” For example, Graph Attention Network (GAT) is a frequently used GNN architecture, which adopts an attention mechanism to carefully select the neighborhood nodes for message passing and aggregation. However, it is difficult to explain why certain neighbors are selected while others are not and how the selected neighbors contribute to the final classification result. In this paper, we present a graph learning model called Explainable Graph Attention Network (XGAT), which integrates graph attention modeling and explainability. We use a single model to target both the accuracy and explainability of problem spaces and show that in the context of graph attention modeling, we can design a unified neighborhood selection strategy that selects appropriate neighbor nodes for both better accuracy and enhanced explainability. To justify this, we conduct extensive experiments to better understand the behavior of our model under different conditions and show an increase in both accuracy and explainability.

Keywords: explainable AI, graph attention network, graph neural network, node classification

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Authors: Khidir R. Sharaf, Didar A. Ali

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The nullity η (G) of a graph is the occurrence of zero as an eigenvalue in its spectra. A zero-sum weighting of a graph G is real valued function, say f from vertices of G to the set of real numbers, provided that for each vertex of G the summation of the weights f (w) over all neighborhood w of v is zero for each v in G.A high zero-sum weighting of G is one that uses maximum number of non-zero independent variables. If G is graph with an end vertex, and if H is an induced sub-graph of G obtained by deleting this vertex together with the vertex adjacent to it, then, η(G)= η(H). In this paper, a high zero-sum weighting technique and the end vertex procedure are applied to evaluate the nullity of t-tupple and generalized t-tupple graphs are derived and determined for some special types of graphs. Also, we introduce and prove some important results about the t-tupple coalescence, Cartesian and Kronecker products of nut graphs.

Keywords: graph theory, graph spectra, nullity of graphs, statistic

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Authors: Khashayar Kashani Jou, Ilnaz Fathololoomi

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Nowadays, access to sustainable development in cities is assumed as one of the most important goals of urban managers. In the meanwhile, neighborhood as the smallest unit of urban spatial organization has a substantial effect on urban sustainability. Hence, attention to and focus on this subject is highly important in urban development plans. The objective of this study is evaluation of the status of Jahanshahr Neighborhood in Karaj city based on sustainable neighborhood development indicators. This research has been applied based on documentary method and field surveys. Also, evaluating of Jahanshahr Neighborhood of Karaj shows that it has a high level in sustainability in physical and economical dimension while a low level in cultural and social dimension. For this purpose, this neighborhood as a semi-sustainable neighborhood must take measures for development of collective spaces and efficiency of utilizing the public neighborhood spaces via collaboration of citizens and officials.

Keywords: neighborhood, sustainable development, sustainable neighborhood development, Jahanshahr neighborhood

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Authors: Zhifei Hu, Feng Xia

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In recent years, Graph neural network has been widely used in knowledge graph recommendation. The existing recommendation methods based on graph neural network extract information from knowledge graph through entity and relation, which may not be efficient in the way of information extraction. In order to better propose useful entity information for the current recommendation task in the knowledge graph, we propose an end-to-end Neural network Model based on multi-stream graph attentional Mechanism (MSGAT), which can effectively integrate the knowledge graph into the recommendation system by evaluating the importance of entities from both users and items. Specifically, we use the attention mechanism from the user's perspective to distil the domain nodes information of the predicted item in the knowledge graph, to enhance the user's information on items, and generate the feature representation of the predicted item. Due to user history, click items can reflect the user's interest distribution, we propose a multi-stream attention mechanism, based on the user's preference for entities and relationships, and the similarity between items to be predicted and entities, aggregate user history click item's neighborhood entity information in the knowledge graph and generate the user's feature representation. We evaluate our model on three real recommendation datasets: Movielens-1M (ML-1M), LFM-1B 2015 (LFM-1B), and Amazon-Book (AZ-book). Experimental results show that compared with the most advanced models, our proposed model can better capture the entity information in the knowledge graph, which proves the validity and accuracy of the model.

Keywords: graph attention network, knowledge graph, recommendation, information propagation

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Authors: Prateek Chharia, Biswa Bhusan Ghosh

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Graph coloring is one of the prominent concepts in graph coloring. It can be defined as a coloring of the various regions of the graph such that all the constraints are fulfilled. In this paper various graphs coloring approaches like greedy coloring, Heuristic search for maximum independent set and graph coloring using edge table is described. Graph coloring can be used in various real time applications like student time tabling generation, Sudoku as a graph coloring problem, GSM phone network.

Keywords: graph coloring, greedy coloring, heuristic search, edge table, sudoku as a graph coloring problem

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Authors: Zahira Anane, Abdelhafid Bayadi

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Corona in the HV overhead transmission lines is an important source of attenuation and distortion of overvoltage surges. This phenomenon of distortion, which is superimposed on the distortion by skin effect, is due to the dissipation of energy by injection of space charges around the conductor, this process with place as soon as the instantaneous voltage exceeds the threshold voltage of the corona effect conductors. This paper presents a mathematical model to determine the corona inception voltage, the critical electric field and the corona radius, to predict the capacitive changes at conductor of transmission line due to corona. This model has been incorporated into the Alternative Transients Program version of the Electromagnetic Transients Program (ATP/EMTP) as a user defined component, using the MODELS interface with NORTON TYPE94 of this program and using the foreign subroutine. For obtained the displacement of corona charge hell, dichotomy mathematical method is used for this computation. The present corona model can be used for computing of distortion and attenuation of transient overvoltage waves being propagated in a transmission line of the very high voltage electric power.

Keywords: high voltage, corona, Type94 NORTON, dichotomy, ATP/EMTP, MODELS, distortion, foreign model

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Authors: Mehmet Hakan Karaata

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In this paper, we first coin a new graph theocratic problem with numerous applications. Second, we provide two algorithms for the problem. The first solution is using a brute-force techniques, whereas the second solution is based on an initial identification of the cycles in the given graph. We then provide a correctness proof of the algorithm. The applications of the problem include graph analysis, graph drawing and network structuring.

Keywords: algorithm, cycle, graph algorithm, graph theory, network structuring

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Authors: Severino Gervacio, Velimor Almonte, Emmanuel Natalio

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A simple graph is planar if it there is a way of drawing it in the plane without edge crossings. A planar graph which is not a proper spanning subgraph of another planar graph is a maximal planar graph. We prove that for complete tripartite graphs of order at most 9, the only ones that contain a spanning maximal planar subgraph are K1,1,1, K2,2,2, K2,3,3, and K3,3,3. The main result gives a necessary and sufficient condition for the complete tripartite graph Kx,y,z to contain a spanning maximal planar subgraph.

Keywords: complete tripartite graph, graph, maximal planar graph, planar graph, subgraph

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Authors: Nileshkumar Vaishnav, Aditya Tatu

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An algebraic framework for processing graph signals axiomatically designates the graph adjacency matrix as the shift operator. In this setup, we often encounter a problem wherein we know the filtered output and the filter coefficients, and need to find out the input graph signal. Solution to this problem using direct approach requires O(N3) operations, where N is the number of vertices in graph. In this paper, we adapt the spectral graph partitioning method for partitioning of graphs and use it to reduce the computational cost of the filtering problem. We use the example of denoising of the temperature data to illustrate the efficacy of the approach.

Keywords: graph signal processing, graph partitioning, inverse filtering on graphs, algebraic signal processing

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Authors: Lin Zhang

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Neighborhood redevelopment is vital to improve residents’ living environment, and there has been a national neighborhood micro-redevelopment initiative in China since 2020, which is largely different from the previous large-scale demolition and reconstruction projects. Yet, few studies systematically examine the new interactions of multiple actors in this initiative. China’s neighborhood (micro-) redevelopment is a kind of governance network, and the complexity perspective could reflect the dynamic nature of multiple actors and their relationships in governance networks. In order to better understand the fundamental shifts of governance networks in China’s neighborhood micro-redevelopment, this paper adopted a theoretical framework of complexity in governance networks and analyzed the new governance networks of neighborhood micro-redevelopment projects in Haikou accordingly.

Keywords: neighborhood redevelopment, governance, networks, Haikou

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Authors: Saieed Akbari, Yousef Bagheri, Amir Hossein Ghodrati, Sima Saadat Akhtar

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Let G be a simple graph with the vertex set V (G) and with the adjacency matrix A (G). The energy E (G) of G is defined to be the sum of the absolute values of all eigenvalues of A (G). Also let n and m be number of edges and vertices of the graph respectively. A regular graph is a graph where each vertex has the same number of neighbours. Given a graph G, its line graph L(G) is a graph such that each vertex of L(G) represents an edge of G; and two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G. In this paper we show that for every regular graphs and also for every line graphs such that (G) 3 we have, E(G) 2nm + n 1. Also at the other part of the paper we prove that 2 (G) E(G) for an arbitrary graph G.

Keywords: eigenvalues, energy, line graphs, matching number

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Authors: Nileshkumar Vishnav, Aditya Tatu

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A recent approach of representing graph signals and graph filters as polynomials is useful for graph signal processing. In this approach, the adjacency matrix plays pivotal role; instead of the more common approach involving graph-Laplacian. In this work, we follow the adjacency matrix based approach and corresponding algebraic signal model. We further expand the theory and introduce the concept of similarity of two graphs. The similarity of graphs is useful in that key properties (such as filter-response, algebra related to graph) get transferred from one graph to another. We demonstrate potential applications of the relation between two similar graphs, such as nonuniform filter design, DTMF detection and signal reconstruction.

Keywords: graph signal processing, algebraic signal processing, graph similarity, isospectral graphs, nonuniform signal processing

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Authors: Bing Li, Zhi Li, Yilong Yang

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Web classification can promote the quality of service discovery and management in the service repository. It is widely used to locate developers desired services. Although traditional classification methods based on supervised learning models can achieve classification tasks, developers need to manually mark web services, and the quality of these tags may not be enough to establish an accurate classifier for service classification. With the doubling of the number of web services, the manual tagging method has become unrealistic. In recent years, the attention mechanism has made remarkable progress in the field of deep learning, and its huge potential has been fully demonstrated in various fields. This paper designs a multi-head attention graph convolutional network (MHAGCN) service classification method, which can assign different weights to the neighborhood nodes without complicated matrix operations or relying on understanding the entire graph structure. The framework combines the advantages of the attention mechanism and graph convolutional neural network. It can classify web services through automatic feature extraction. The comprehensive experimental results on a real dataset not only show the superior performance of the proposed model over the existing models but also demonstrate its potentially good interpretability for graph analysis.

Keywords: attention mechanism, graph convolutional network, interpretability, service classification, service discovery

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Authors: M. Hussain, Aqsa Farooq

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Let G = (V,E) be a connected graph and distance between any two vertices a and b in G is a−b geodesic and is denoted by d(a, b). A set of vertices W resolves a graph G if each vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G. In this paper line graph of honeycomb network has been derived and then we calculated the metric dimension on line graph of honeycomb network.

Keywords: Resolving set, Metric dimension, Honeycomb network, Line graph

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Authors: Xiaoheng Tan, Xianfang Li, Tan Guo, Yuchuan Liu, Zhijun Yang, Hongye Li, Kai Fu, Yufang Wu, Heling Gong

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The image data collected in reality often have high dimensions, and it contains noise and redundant information. Therefore, it is necessary to extract the compact feature expression of the original perceived image. In this process, effective use of prior knowledge such as data structure distribution and sample label is the key to enhance image feature discrimination and robustness. Based on the above considerations, this paper proposes a local preserving discriminant feature learning model based on graph optimization. The model has the following characteristics: (1) Locality preserving constraint can effectively excavate and preserve the local structural relationship between data. (2) The flexibility of graph learning can be improved by constructing a new local geometric structure graph using label information and the nearest neighbor threshold. (3) The L₂,₁ norm is used to redefine LDA, and the diagonal matrix is introduced as the scale factor of LDA, and the samples are selected, which improves the robustness of feature learning. The validity and robustness of the proposed algorithm are verified by experiments in two public image datasets.

Keywords: feature extraction, graph optimization local preserving projection, linear discriminant analysis, L₂, ₁ norm

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Authors: Qiuyi Lyu, Bin Gong

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Breadth-First Search(BFS) is a core graph algorithm that is widely used for graph analysis. As it is frequently used in many graph applications, improve the BFS performance is essential. In this paper, we present a graph ordering method that could reorder the graph nodes to achieve better data locality, thus, improving the BFS performance. Our method is based on an observation that the sibling relationships will dominate the cache access pattern during the BFS traversal. Therefore, we propose a frequency-based model to construct the graph order. First, we optimize the graph order according to the nodes’ visit frequency. Nodes with high visit frequency will be processed in priority. Second, we try to maximize the child nodes overlap layer by layer. As it is proved to be NP-hard, we propose a heuristic method that could greatly reduce the preprocessing overheads. We conduct extensive experiments on 16 real-world datasets. The result shows that our method could achieve comparable performance with the state-of-the-art methods while the graph ordering overheads are only about 1/15.

Keywords: breadth-first search, BFS, graph ordering, graph algorithm

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Authors: Lijuan Zhou, Mengqi Wu, Changyong Niu

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Attributed graph clustering can utilize the graph topology and node attributes to uncover hidden community structures and patterns in complex networks, aiding in the understanding and analysis of complex systems. Utilizing contrastive learning for attributed graph clustering can effectively exploit meaningful implicit relationships between data. However, existing attributed graph clustering methods based on contrastive learning suffer from the following drawbacks: 1) Complex data augmentation increases computational cost, and inappropriate data augmentation may lead to semantic drift. 2) The selection of positive and negative samples neglects the intrinsic cluster structure learned from graph topology and node attributes. Therefore, this paper proposes a method called self-supervised Attributed Graph Clustering with Dual Contrastive Loss constraints (AGC-DCL). Firstly, Siamese Multilayer Perceptron (MLP) encoders are employed to generate two views separately to avoid complex data augmentation. Secondly, the neighborhood contrastive loss is introduced to constrain node representation using local topological structure while effectively embedding attribute information through attribute reconstruction. Additionally, clustering-oriented contrastive loss is applied to fully utilize clustering information in global semantics for discriminative node representations, regarding the cluster centers from two views as negative samples to fully leverage effective clustering information from different views. Comparative clustering results with existing attributed graph clustering algorithms on six datasets demonstrate the superiority of the proposed method.

Keywords: attributed graph clustering, contrastive learning, clustering-oriented, self-supervised learning

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Authors: S. R. Nayaka, Putta Swamy

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Graph polynomial is one of the algebraic representations of the Graph. The degree polynomial is one of the simple algebraic representations of graphs. The degree polynomial of a graph G of order n is the polynomial Deg(G, x) with the coefficients deg(G,i) where deg(G,i) denotes the number of vertices of degree i in G. In this article, we investigate the behavior of the roots of some families of Graphs in the complex field. We investigate for the graphs having only integral roots. Further, we characterize the graphs having single roots or having real roots and behavior of the polynomial at the particular value is also obtained.

Keywords: degree polynomial, regular graph, minimum and maximum degree, graph operations

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Authors: Ladznar S. Laja, Rosalio G. Artes, Jr.

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This paper introduced a graph polynomial relating convexity concepts. A graph polynomial is a polynomial representing a graph given some parameters. On the other hand, a subgraph H of a graph G is said to be convex in G if for every pair of vertices in H, every shortest path with these end-vertices lies entirely in H. We define the convex subgraph polynomial of a graph G to be the generating function of the sequence of the numbers of convex subgraphs of G of cardinalities ranging from zero to the order of G. This graph polynomial is monic since G itself is convex. The convex index which counts the number of convex subgraphs of G of all orders is just the evaluation of this polynomial at 1. Relationships relating algebraic properties of convex subgraphs polynomial with graph theoretic concepts are established.

Keywords: convex subgraph, convex index, generating function, polynomial ring

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Authors: Samai'la Abdullahi

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A graph is a pair of two set and so that a graph is a pictorial representation of a system using two basic element nodes and edges. A node is represented by a circle (either hallo shade) and edge is represented by a line segment connecting two nodes together. In this paper, we present a circuit network in the concept of graph theory application and also circuit models of graph are represented in logical connection method were we formulate matrix method of adjacency and incidence of matrix and application of truth table.

Keywords: euler circuit and path, graph representation of circuit networks, representation of graph models, representation of circuit network using logical truth table

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Authors: A. Lemzadmi

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Rotational and vibrational temperatures of the SF6-N2 gas mixture are spectroscopically measured over a pressure range of 2-14 bars. The spectra obtained of the light emission of the corona discharge were recorded with different values of pressure, voltage and current together with the variation of the position of the tip electrode. The emission of N2 is very dominant for different gas concentration and the second positive system 2S+ is the most important. The convolution method is used for the determination of the temperature. The Rotational temperature measurements of the plasma reveal gas temperatures in the range of 450-650°K and vibrational temperatures in the range of 1800-2200°K.

Keywords: rotational temperatures, corona discharges, SF6-N2 gas mixture, vibrational temperatures

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Authors: A. H. El-Shazly, A. El-Tayeb, M. F. Elkady, Mona G. E. Ibrahim, Abdelazim M. Negm

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In this paper, corona discharge plasma reactor was used for degradation of organic pollution in aqueous solutions in batch reactor. This work examines the possibility of increasing the organic pollution removal efficiency from wastewater using non-thermal plasma. Three types of organic pollution phenol, acid blue 25 and methylene blue are presented to investigate experimentally the amount of organic pollution removal efficiency from wastewater. Measurement results for phenol degradation percentage are 71% in 35 min and 96% when its residence time is 60 min. In addition, the degradation behavior of acid blue 25 utilizing dual pin-to-plate corona discharge plasma system displays a removal efficiency of 82% in 11 min. The complete decolorization was accomplished in 35 min for concentration of acid blue 25 up to 100 ppm. Furthermore, the methylene blue degradation touched up to 85% during 35 min treatment in corona discharge plasma a batch reactor system. The decolorization ratio, conductivity, corona current and discharge energy are considered at various concentration molarity for AlCl3, CaCl2, KCl and NaCl under different molar concentration. It was observed that the attendance of salts at the same concentration level considerably diminished the rate and the extent of decolorization. The research presented that the corona system could be positively utilized in a diversity of organically contaminated at diverse concentrations. Energy consumption requirements for decolorization was considered. The consequences will be valuable for designing the plasma treatment systems appropriate for industrial wastewaters.

Keywords: wastewater treatment, corona discharge, non-thermal plasma, organic pollution

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Authors: Krishnendra Shekhawat

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Given a graph G = (V, E), a rectangular dual of G represents the vertices of G by a set of interior-disjoint rectangles such that two rectangles touch if and only if there is an edge between the two corresponding vertices in G. Rectangular duals do not exist for every graph, so we can define maximal rectangular duals. A maximal rectangular dual is a rectangular dual of a graph G such that there exists no graph G ′ with a rectangular dual where G is a subgraph of G ′. In this paper, we enumerate all maximal rectangular duals (or, to be precise, the corresponding planar graphs) up to six nodes and presents a necessary condition for the existence of a rectangular dual. This work allegedly has applications in integrated circuit design and architectural floor plans.

Keywords: adjacency, degree sequence, dual graph, rectangular dual

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Authors: Asraful Islam

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Extended dependency graph (EDG) is a state-of-the-art isomorphic graph to represent normal logic programs (NLPs) that can characterize the consistency of NLPs by graph analysis. To construct the vertices and arcs of an EDG, additional renaming atoms and rules besides those the given program provides are used, resulting in higher space complexity compared to the corresponding traditional dependency graph (TDG). In this article, we propose an extended labeled dependency graph (ELDG) to represent an NLP that shares an equal number of nodes and arcs with TDG and prove that it is isomorphic to the domain program. The number of nodes and arcs used in the underlying dependency graphs are formulated to compare the space complexity. Results show that ELDG uses less memory to store nodes, arcs, and cycles compared to EDG. To exhibit the desirability of ELDG, firstly, the stable models of the kernel form of NLP are characterized by the admissible coloring of ELDG; secondly, a relation of the stable models of a kernel program with the handles of the minimal, odd cycles appearing in the corresponding ELDG has been established; thirdly, to our best knowledge, for the first time an inverse transformation from a dependency graph to the representing NLP w.r.t. ELDG has been defined that enables transferring analytical results from the graph to the program straightforwardly.

Keywords: normal logic program, isomorphism of graph, extended labelled dependency graph, inverse graph transforma-tion, graph colouring

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Authors: Boumesbah Asma, Chergui Mohamed El-amine

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Since it has been shown that the multi-objective minimum spanning tree problem (MOST) is NP-hard even with two criteria, we propose in this study a hybrid NSGA-II algorithm with an exact mutation operator, which is only used with low probability, to find an approximation to the Pareto front of the problem. In a connected graph G, a spanning tree T of G being a connected and cycle-free graph, if k edges of G\T are added to T, we obtain a partial graph H of G inducing a reduced size multi-objective spanning tree problem compared to the initial one. With a weak probability for the mutation operator, an exact method for solving the reduced MOST problem considering the graph H is then used to give birth to several mutated solutions from a spanning tree T. Then, the selection operator of NSGA-II is activated to obtain the Pareto front approximation. Finally, an adaptation of the VNS metaheuristic is called for further improvements on this front. It allows finding good individuals to counterbalance the diversification and the intensification during the optimization search process. Experimental comparison studies with an exact method show promising results and indicate that the proposed algorithm is efficient.

Keywords: minimum spanning tree, multiple objective linear optimization, combinatorial optimization, non-sorting genetic algorithm, variable neighborhood search

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