Search results for: cartesian graphs

Authors: Verónica Díaz

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A Cartesian graph, as a mathematical object, becomes a tool for configuration of change. Its best comprehension is done through everyday life problem-solving associated with its representation. Despite this, the current educational framework favors general graphs, without consideration of their argumentation. Students are required to find the mathematical function without associating it to the development of graphical language. This research describes the use made by students of configurations made prior to Cartesian graphs with regards to an everyday life problem related to a time and distance variation phenomenon. The theoretical framework describes the function conditions of study and their modeling. This is a qualitative, descriptive study involving six undergraduate case studies that were carried out during the first term in 2016 at University of Los Lagos. The research problem concerned the graphic modeling of a real person’s movement phenomenon, and two levels of analysis were identified. The first level aims to identify local and global graph interpretations; a second level describes the iconicity and referentiality degree of an image. According to the results, students were able to draw no figures before the Cartesian graph, highlighting the need for students to represent the context and the movement of which causes the phenomenon change. From this, they managed Cartesian graphs representing changes in position, therefore, achieved an overall view of the graph. However, the local view only indicates specific events in the problem situation, using graphic and verbal expressions to represent movement. This view does not enable us to identify what happens on the graph when the movement characteristics change based on possible paths in the person’s walking speed.

Keywords: cartesian graphs, higher education, movement modeling, problem solving

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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: 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: Kaan Karaoglu, Raif Bayir

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In this study, a cartesian robot is developed to improve the performance and efficiency of wireless charging of electric vehicles. The cartesian robot has three axes, each of which moves linearly. Magnetic positioning is used to align the cartesian robot transmitter charging pad. There are two different wireless charging methods, static and dynamic, for charging electric vehicles. The current state of charge information (SOC State of Charge) and location information are received wirelessly from the electric vehicle. Based on this information, the power to be transmitted is determined, and the transmitter and receiver charging pads are aligned for maximum efficiency. With this study, a fully automated cartesian robot structure will be used to charge electric vehicles with the highest possible efficiency. With the wireless communication established between the electric vehicle and the charging station, the charging status will be monitored in real-time. The cartesian robot developed in this study is a fully automatic system that can be easily used in static wireless charging systems with vehicle-machine communication.

Keywords: electric vehicle, wireless charging systems, energy efficiency, cartesian robot, location detection, trajectory planning

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Authors: Gete Umbrey, Saifur Rahman

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Graphs are found to be most capable in computing, and its abstract structures have been applied in some specific computations and algorithms like in phase encoding controller, processor microcontroller, and synthesis of a CMOS switching network, etc. Being motivated by these works, we develop an independent approach to study semiring structures and various properties by defining the binary operations which in fact, seems analogous to an existing definition in some sense but with a different approach. This work emphasizes specifically on the construction of semigroup and semiring structures on the set of undirected graphs, and their properties are investigated therein. It is expected that the investigation done here may have some interesting applications in theoretical computer science, networking and decision making, and also on joining of two network systems.

Keywords: graphs, join and union of graphs, semiring, weighted graphs

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Authors: Seyed Ahmad Mojallal

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Let G be a connected threshold graph of order n with m edges and trace T. In this talk we give a lower bound on Laplacian energy in terms of n, m, and T of G. From this we determine the threshold graphs with the first four minimal Laplacian energies. We also list the first 20 minimal Laplacian energies among threshold graphs. Let σ=σ(G) be the number of Laplacian eigenvalues greater than or equal to average degree of graph G. Using this concept, we obtain the threshold graphs with the largest and the second largest Laplacian energies.

Keywords: Laplacian eigenvalues, Laplacian energy, threshold graphs, extremal graphs

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Authors: Y. Wang

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The Traveling salesman problem (TSP) is NP-hard in combinatorial optimization. The research shows the algorithms for TSP on the sparse graphs have the shorter computation time than those for TSP according to the complete graphs. We present an improved iterative algorithm to compute the sparse graphs for TSP by frequency graphs computed with frequency quadrilaterals. The iterative algorithm is enhanced by adjusting two parameters of the algorithm. The computation time of the algorithm is O(CN_maxn²) where C is the iterations, N_max is the maximum number of frequency quadrilaterals containing each edge and n is the scale of TSP. The experimental results showed the computed sparse graphs generally have less than 5n edges for most of these Euclidean instances. Moreover, the maximum degree and minimum degree of the vertices in the sparse graphs do not have much difference. Thus, the computation time of the methods to resolve the TSP on these sparse graphs will be greatly reduced.

Keywords: frequency quadrilateral, iterative algorithm, sparse graph, traveling salesman problem

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Authors: T. Pathinathan, M. Peter

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Fuzzy graphs incorporate concepts from graph theory with fuzzy principles. In this paper, we make a study on the properties of fuzzy graphs which are non-cyclic and are of two-dimensional in structure. In particular, this paper presents 2D structure or the structure of double layer for a non-cyclic fuzzy graph whose underlying crisp graph is non-cyclic. In any graph structure, introducing 2D structure may lead to an inherent cycle. We propose relevant conditions for 2D structured non-cyclic fuzzy graphs. These conditions are extended even to fuzzy graphs of the 3D structure. General theoretical properties that are studied for any fuzzy graph are verified to 2D structured or double layered fuzzy graphs. Concepts like Order, Degree, Strong and Size for a fuzzy graph are studied for 2D structured or double layered non-cyclic fuzzy graphs. Using different types of fuzzy graphs, the proposed concepts relating to 2D structured fuzzy graphs are verified.

Keywords: double layered fuzzy graph, double layered non–cyclic fuzzy graph, order, degree and size

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Authors: Swapnil Gupta, C. Pandu Rangan

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A uniquely restricted matching is defined to be a matching M whose matched vertices induces a sub-graph which has only one perfect matching. In this paper, we make progress on the open question of the status of this problem on interval graphs (graphs obtained as the intersection graph of intervals on a line). We give an algorithm to compute maximum cardinality uniquely restricted matchings on certain sub-classes of interval graphs. We consider two sub-classes of interval graphs, the former contained in the latter, and give O(|E|^2) time algorithms for both of them. It is to be noted that both sub-classes are incomparable to proper interval graphs (graphs obtained as the intersection graph of intervals in which no interval completely contains another interval), on which the problem can be solved in polynomial time.

Keywords: uniquely restricted matching, interval graph, matching, induced matching, witness counting

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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: Robert Gold

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Control flow graphs are a well-known representation of the sequential control flow structure of programs with a multitude of applications. Not only single functions but also sets of functions or complete programs can be modelled by control flow graphs. In this case the size of the graphs can grow considerably and thus makes it difficult for software engineers to analyse the control flow. Graph reductions are helpful in this situation. In this paper we define reductions to subsets of nodes. Since executions of programs are represented by paths through the control flow graphs, paths should be preserved. Furthermore, the composition of reductions makes a stepwise analysis approach possible.

Keywords: control flow graph, graph reduction, software engineering, software applications

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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: Josef Slapal

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Digital images are approximations of real ones and, therefore, to be able to study them, we need the digital plane Z2 to be equipped with a convenient structure that behaves analogously to the Euclidean topology on the real plane. In particular, it is required that such a structure allows for a digital analogue of the Jordan curve theorem. We introduce certain adjacency graphs on the digital plane and prove digital Jordan curves for them thus showing that the graphs provide convenient structures on Z2 for the study and processing of digital images. Further convenient structures including the wellknown Khalimsky and Marcus-Wyse adjacency graphs may be obtained as quotients of the graphs introduced. Since digital Jordan curves represent borders of objects in digital images, the adjacency graphs discussed may be used as background structures on the digital plane for solving the problems of digital image processing that are closely related to borders like border detection, contour filling, pattern recognition, thinning, etc.

Keywords: digital plane, adjacency graph, Jordan curve, quotient adjacency

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Authors: Shin-Shin Kao, Yuan-Kang Shih, Hsun Su

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In this paper, we show that the conjecture of Chv tal, which states that any 1-tough graph is either a Hamiltonian graph or its complement contains a specific graph denoted by F, does not hold in general. More precisely, it is true only for graphs with six or seven vertices, and is false for graphs with eight or more vertices. A theorem is derived as a correction for the conjecture.

Keywords: complement, degree sum, hamiltonian, tough

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Authors: Zahra Khalid, Gul Muhammad Khan, Arbab Masood Ahmad

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Cartesian Genetic Programming (CGP) is explored to design an optimal circuit capable of early stage breast cancer detection. CGP is used to evolve simple multiplexer circuits for detection of malignancy in the Fine Needle Aspiration (FNA) samples of breast. The data set used is extracted from Wisconsins Breast Cancer Database (WBCD). A range of experiments were performed, each with different set of network parameters. The best evolved network detected malignancy with an accuracy of 99.14%, which is higher than that produced with most of the contemporary non-linear techniques that are computational expensive than the proposed system. The evolved network comprises of simple multiplexers and can be implemented easily in hardware without any further complications or inaccuracy, being the digital circuit.

Keywords: breast cancer detection, cartesian genetic programming, evolvable hardware, fine needle aspiration

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Authors: Yuchen Yang, Zhenming Wang, Jun Zhu, Ning Zhao

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An easy-to-implement and robust finite volume multi-resolution Weighted Essentially Non-Oscillatory (WENO) scheme is proposed on adaptive cartesian grids in this paper. Such a multi-resolution WENO scheme is combined with the ghost cell immersed boundary method (IBM) and wall-function technique to solve Navier-Stokes equations. Unlike the k-exact finite volume WENO schemes which involve large amounts of extra storage, repeatedly solving the matrix generated in a least-square method or the process of calculating optimal linear weights on adaptive cartesian grids, the present methodology only adds very small overhead and can be easily implemented in existing edge-based computational fluid dynamics (CFD) codes with minor modifications. Also, the linear weights of this adaptive finite volume multi-resolution WENO scheme can be any positive numbers on condition that their sum is one. It is a way of bypassing the calculation of the optimal linear weights and such a multi-resolution WENO scheme avoids dealing with the negative linear weights on adaptive cartesian grids. Some benchmark viscous problems are numerical solved to show the efficiency and good performance of this adaptive multi-resolution WENO scheme. Compared with a second-order edge-based method, the presented method can be implemented into an adaptive cartesian grid with slight modification for big Reynolds number problems.

Keywords: adaptive mesh refinement method, finite volume multi-resolution WENO scheme, immersed boundary method, wall-function technique.

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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: Bin Sheng, Changhong Lu

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Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by γₚᵣ(G), is the size of its smallest paired dominating set. Goddard and Henning conjectured that γₚᵣ(G) ≤ 4n/7 holds for every graph G with δ(G) ≥ 3, except the Petersen Graph. In this paper, we prove this conjecture for cubic graphs.

Keywords: paired dominating set, upper bound, cubic graphs, weight function

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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: Zhengkun Guo, Yanbin Li, Wenqing Wang, Bo Zou

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The statically fused converted position and doppler measurements Kalman filter (SF-CMKF) with additive debiased measurement conversion has been previously presented to combine the resulting states of converted position measurements Kalman filter (CPMKF) and converted doppler measurement Kalman filter (CDMKF) to yield the final state estimates under minimum mean squared error (MMSE) criterion. However, the exact compensation for the bias in the polar-to-cartesian and spherical-to-cartesian conversion are multiplicative and depend on the statistics of the cosine of the angle measurement errors. As a result, the consistency and performance of the SF-CMKF may be suboptimal in large-angle error situations. In this paper, the multiplicative unbiased position and Doppler measurement conversion for 2D (polar-to-cartesian) tracking are derived, and the SF-CMKF is improved to use those conversions. Monte Carlo simulations are presented to demonstrate the statistical consistency of the multiplicative unbiased conversion and the superior performance of the modified SF-CMKF (SF-UCMKF).

Keywords: measurement conversion, Doppler, Kalman filter, estimation, tracking

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Authors: Abdul Jalil M. Khalaf, Esraa M. Kadhim

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The Hosoya polynomial of a graph G is a graphical invariant polynomial that its first derivative at x= 1 is equal to the Wiener index and second derivative at x=1 is equal to the Hyper-Wiener index. In this paper we study the Hosoya polynomial of zero-divisor graphs.

Keywords: Hosoya polynomial, wiener index, Hyper-Wiener index, zero-divisor graphs

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Authors: Ricardo Merlo

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In this paper, we analyzed the different didactic proposals for teaching about the free fall motion of bodies available online. An important aspect was the interpretation of the direction and sense of the acceleration of gravity and of the falling velocity of a body, which is why we found different applications of the Cartesian reference system used and also different graphical presentations of the velocity as a function of time and of the distance traveled vertically by the body in the period of time that it was dropped from a height h0. In this framework, a survey of previous concepts was applied to a voluntary group of first-year university students of an Engineering degree before and after the development of the class of the subject in question. Then, Hake's index (0.52) was determined, which resulted in an average learning gain from the meaningful use of the reference system and the respective graphs of v=ƒ (t) and h=ƒ (t).

Keywords: didactic gain, free–fall, physics teaching, previous knowledge

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Authors: Ivan Belik

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In terms of social networks split graphs correspond to the variety of interpersonal and intergroup relations. In this paper we analyse the interaction between the cliques (socially strong and trusty groups) and the independent sets (fragmented and non-connected groups of people) as the basic components of any split graph. Based on the Semi-Lagrangean relaxation for the k-cardinality assignment problem we show the way of how to minimize the socially risky interactions between the cliques and the independent sets within the social network.

Keywords: cliques, independent sets, k-cardinality assignment, social networks, split graphs

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Authors: Amin Amini

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A Kinect sensor has been utilized as a cheap and accurate alternative to 3D laser scanners and electronic distance measurement (EDM) systems. This research presents an inexpensive and easy-to-setup system that utilizes the Microsoft Kinect v2 sensor as a surveying and measurement tool and investigates the possibility of using such a device as a replacement for conventional theodolite systems. The system was tested in an indoor environment where its accuracy in distance and angle measurements was tested using virtual markers in a 3D Cartesian environment. The system has shown an average accuracy of 97.94 % in measuring distances and 99.11 % and 98.84 % accuracy for area and perimeter, respectively, within the Kinect’s surveying range of 1.5 to 6 meters. The research also tested the system competency for relative angle determination between two objects.

Keywords: kinect v2, 3D measurement, depth map, ToF

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Authors: Sanju Vaidya, Aihua Li

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Vulnerability measures and topological indices are crucial in solving various problems such as the stability of the communication networks and development of mathematical models for chemical compounds. In 1947, Harry Wiener introduced a topological index related to molecular branching. Now there are more than 100 topological indices for graphs. For example, Hosoya polynomials (also called Wiener polynomials) were introduced to derive formulas for certain vulnerability measures and topological indices for various graphs. In this paper, we will find a relation between the Hosoya polynomials of any graph and its Mycielskian graph. Additionally, using this we will compute vulnerability measures, closeness and betweenness centrality, and extended Wiener indices. It is fascinating to see how Hosoya polynomials are useful in the two diverse fields, cybersecurity and chemistry.

Keywords: hosoya polynomial, mycielskian graph, graph vulnerability measure, topological index

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Authors: Analen Malnegro, Gina Malacas

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Dominating sets and related topics have been studied extensively in the past few decades. A dominating set of a graph G is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G. Some problems involving detection devices can be modeled with graphs. Finding the minimum number of devices needed according to the type of devices and the necessity of locating the object gives rise to locating-dominating sets. A subset S of vertices of a graph G is called locating-dominating set, LD-set for short, if it is a dominating set and if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. The location-domination number λ(G) is the minimum cardinality of an LD-set for G. The complement of a graph G is a graph Ḡ on same vertices such that two distinct vertices of Ḡ are adjacent if and only if they are not adjacent in G. An LD-set of a graph G is global if it is an LD-set of both G and its complement Ḡ. The global location-domination number λg(G) is defined as the minimum cardinality of a global LD-set of G. In this paper, global LD-sets on the join of two graphs are characterized. Global location-domination numbers of these graphs are also determined.

Keywords: dominating set, global locating-dominating set, global location-domination number, locating-dominating set, location-domination number

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Authors: Ali Ateş, Ansar B. Mwimbo, Ali H. Abdulkarim

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General transport equation has a wide range of application in Fluid Mechanics and Heat Transfer problems. In this equation, generally when φ variable which represents a flow property is used to represent fluid velocity component, general transport equation turns into momentum equations or with its well known name Navier-Stokes equations. In these non-linear differential equations instead of seeking for analytic solutions, preferring numerical solutions is a more frequently used procedure. Finite difference method is a commonly used numerical solution method. In these equations using velocity and pressure gradients instead of stress tensors decreases the number of unknowns. Also, continuity equation, by integrating the system, number of equations is obtained as number of unknowns. In this situation, velocity and pressure components emerge as two important parameters. In the solution of differential equation system, velocities and pressures must be solved together. However, in the considered grid system, when pressure and velocity values are jointly solved for the same nodal points some problems confront us. To overcome this problem, using staggered grid system is a referred solution method. For the computerized solutions of the staggered grid system various algorithms were developed. From these, two most commonly used are SIMPLE and SIMPLER algorithms. In this study Navier-Stokes equations were numerically solved for Newtonian flow, whose mass or gravitational forces were neglected, for incompressible and laminar fluid, as a hydro dynamically fully developed region and in two dimensional cartesian coordinate system. Finite difference method was chosen as the solution method. This is a parametric study in which varying values of velocity components, pressure and Reynolds numbers were used. Differential equations were discritized using central difference and hybrid scheme. The discritized equation system was solved by Gauss-Siedel iteration method. SIMPLE and SIMPLER were used as solution algorithms. The obtained results, were compared for central difference and hybrid as discritization methods. Also, as solution algorithm, SIMPLE algorithm and SIMPLER algorithm were compared to each other. As a result, it was observed that hybrid discritization method gave better results over a larger area. Furthermore, as computer solution algorithm, besides some disadvantages, it can be said that SIMPLER algorithm is more practical and gave result in short time. For this study, a code was developed in DELPHI programming language. The values obtained in a computer program were converted into graphs and discussed. During sketching, the quality of the graph was increased by adding intermediate values to the obtained result values using Lagrange interpolation formula. For the solution of the system, number of grid and node was found as an estimated. At the same time, to indicate that the obtained results are satisfactory enough, by doing independent analysis from the grid (GCI analysis) for coarse, medium and fine grid system solution domain was obtained. It was observed that when graphs and program outputs were compared with similar studies highly satisfactory results were achieved.

Keywords: finite difference method, GCI analysis, numerical solution of the Navier-Stokes equations, SIMPLE and SIMPLER algoritms

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Authors: Amira Dhokar, Lobna Hlaoua, Lotfi Ben Romdhane

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Tweet contextualization (TC) is a new issue that aims to answer questions of the form 'What is this tweet about?' The idea of this task was imagined as an extension of a previous area called multi-document summarization (MDS), which consists in generating a summary from many sources. In both TC and MDS, the summary should ideally contain the most relevant information of the topic that is being discussed in the source texts (for MDS) and related to the query (for TC). Furthermore of being informative, a summary should be coherent, i.e. well written to be readable and grammatically compact. Hence, coherence is an essential characteristic in order to produce comprehensible texts. In this paper, we propose a new approach to improve readability and coherence for tweet contextualization based on bipartite graphs. The main idea of our proposed method is to reorder sentences in a given paragraph by combining most expressive words detection and HITS (Hyperlink-Induced Topic Search) algorithm to make up a coherent context.

Keywords: bipartite graphs, readability, summarization, tweet contextualization

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Authors: Stefan Wagenpfeil, Felix Engel, Paul McKevitt, Matthias Hemmje

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Multimedia Indexing and Retrieval is generally designed and implemented by employing feature graphs. These graphs typically contain a significant number of nodes and edges to reflect the level of detail in feature detection. A higher level of detail increases the effectiveness of the results but also leads to more complex graph structures. However, graph-traversal-based algorithms for similarity are quite inefficient and computation intensive, especially for large data structures. To deliver fast and effective retrieval, an efficient similarity algorithm, particularly for large graphs, is mandatory. Hence, in this paper, we define a graph-projection into a 2D space (Graph Code) as well as the corresponding algorithms for indexing and retrieval. We show that calculations in this space can be performed more efficiently than graph-traversals due to a simpler processing model and a high level of parallelization. In consequence, we prove that the effectiveness of retrieval also increases substantially, as Graph Codes facilitate more levels of detail in feature fusion. Thus, Graph Codes provide a significant increase in efficiency and effectiveness (especially for Multimedia indexing and retrieval) and can be applied to images, videos, audio, and text information.

Keywords: indexing, retrieval, multimedia, graph algorithm, graph code

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Authors: J. C. Valenzuela-Tripodoro, P. Álvarez-Ruíz, M. A. Mateos-Camacho, M. Cera

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In 2004, it was introduced the concept of Roman domination in graphs. This concept was initially inspired and related to the defensive strategy of the Roman Empire. An undefended place is a city so that no legions are established on it, whereas a strong place is a city in which two legions are deployed. This situation may be modeled by labeling the vertices of a finite simple graph with labels {0, 1, 2}, satisfying the condition that any 0-vertex must be adjacent to, at least, a 2-vertex. Roman domination in graphs is a variant of classic domination. Clearly, the main aim is to obtain such labeling of the vertices of the graph with minimum cost, that is to say, having minimum weight (sum of all vertex labels). Formally, a function f: V (G) → {0, 1, 2} is a Roman dominating function (RDF) in the graph G = (V, E) if f(u) = 0 implies that f(v) = 2 for, at least, a vertex v which is adjacent to u. The weight of an RDF is the positive integer w(f)= ∑_(v∈V)▒〖f(v)〗. The Roman domination number, γ_R (G), is the minimum weight among all the Roman dominating functions? Obviously, the set of vertices with a positive label under an RDF f is a dominating set in the graph, and hence γ(G)≤γ_R (G). In this work, we start the study of a generalization of RDF in which we consider that any undefended place should be defended from a sudden attack by, at least, k legions. These legions can be deployed in the city or in any of its neighbours. A function f: V → {0, 1, . . . , k + 1} such that f(N[u]) ≥ k + |AN(u)| for all vertex u with f(u) < k, where AN(u) represents the set of active neighbours (i.e., with a positive label) of vertex u, is called a [k]-multiple Roman dominating functions and it is denoted by [k]-MRDF. The minimum weight of a [k]-MRDF in the graph G is the [k]-multiple Roman domination number ([k]-MRDN) of G, denoted by γ_[kR] (G). First, we prove that the [k]-multiple Roman domination decision problem is NP-complete even when restricted to bipartite and chordal graphs. A problem that had been resolved for other variants and wanted to be generalized. We know the difficulty of calculating the exact value of the [k]-MRD number, even for families of particular graphs. Here, we present several upper and lower bounds for the [k]-MRD number that permits us to estimate it with as much precision as possible. Finally, some graphs with the exact value of this parameter are characterized.

Keywords: multiple roman domination function, decision problem np-complete, bounds, exact values

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