Threshold-coloring and unit-cube contact representation of planar graphs

In this paper we study threshold-coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. A pair of vertices with a small difference in their colors implies that the edge between them is present, while a pair of vertices with a big color difference implies that the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. Variants of the threshold-coloring problem are related to well-known graph coloring and other graph-theoretic problems. Using these relations we show the NP-completeness for two of these variants, and describe a polynomial-time algorithm for another. © 2015 Elsevier B.V

Threshold-coloring and unit-cube contact representation of graphs

We study threshold coloring of graphs where the vertex colors, represented by integers, describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is present and pairs of vertices with far colors imply the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. We show the NP-completeness for two variants of the threshold coloring problem and describe a polynomial-time algorithm for another. © 2013 Springer-Verlag

Weak unit disk and interval representation of graphs

We study a variant of intersection representations with unit balls: unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far edges, the goal is to represent the vertices of the graph by unit-size balls so that the balls for two adjacent vertices intersect if and only if the corresponding edge is near. We consider the problem in the plane and prove that it is NP-hard to decide whether such a representation exists for a given edgepartition. On the other hand, we show that series-parallel graphs (which include outerplanar graphs) admit such a representation with unit disks for any near/far bipartition of the edges. The unit-interval on the line variant is equivalent to threshold graph coloring, in which context it is known that there exist girth-3 planar graphs (even outerplanar graphs) that do not admit such coloring. We extend this result to girth-4 planar graphs. On the other hand, we show that all triangle-free outerplanar graphs and all planar graphs with maximum average degree less than 26/11 have such a coloring, via unit-interval intersection representation on the line. This gives a simple proof that all planar graphs with girth at least 13 have a unit-interval intersection representation on the line. © Springer International Publishing Switzerland 2016

Happy edges: Threshold-coloring of regular lattices

We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into near and far sets and an integer threshold t, a threshold-coloring of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by t or less, while endpoints of far edges differ by more than t. We study threshold-coloring of tilings of the plane by regular polygons, known as Archimedean lattices, and their duals, the Laves lattices. We prove that some are threshold-colorable with constant number of colors for any edge labeling, some require an unbounded number of colors for specific labelings, and some are not threshold-colorable. © 2014 Springer International Publishing

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

A composable approach to design of newer techniques for large-scale denial-of-service attack attribution

Since its early days, the Internet has witnessed not only a phenomenal growth, but also a large number of security attacks, and in recent years, denial-of-service (DoS) attacks have emerged as one of the top threats. The stateless and destination-oriented Internet routing combined with the ability to harness a large number of compromised machines and the relative ease and low costs of launching such attacks has made this a hard problem to address. Additionally, the myriad requirements of scalability, incremental deployment, adequate user privacy protections, and appropriate economic incentives has further complicated the design of DDoS defense mechanisms. While the many research proposals to date have focussed differently on prevention, mitigation, or traceback of DDoS attacks, the lack of a comprehensive approach satisfying the different design criteria for successful attack attribution is indeed disturbing. Our first contribution here has been the design of a composable data model that has helped us represent the various dimensions of the attack attribution problem, particularly the performance attributes of accuracy, effectiveness, speed and overhead, as orthogonal and mutually independent design considerations. We have then designed custom optimizations along each of these dimensions, and have further integrated them into a single composite model, to provide strong performance guarantees. Thus, the proposed model has given us a single framework that can not only address the individual shortcomings of the various known attack attribution techniques, but also provide a more wholesome counter-measure against DDoS attacks. Our second contribution here has been a concrete implementation based on the proposed composable data model, having adopted a graph-theoretic approach to identify and subsequently stitch together individual edge fragments in the Internet graph to reveal the true routing path of any network data packet. The proposed approach has been analyzed through theoretical and experimental evaluation across multiple metrics, including scalability, incremental deployment, speed and efficiency of the distributed algorithm, and finally the total overhead associated with its deployment. We have thereby shown that it is realistically feasible to provide strong performance and scalability guarantees for Internet-wide attack attribution. Our third contribution here has further advanced the state of the art by directly identifying individual path fragments in the Internet graph, having adopted a distributed divide-and-conquer approach employing simple recurrence relations as individual building blocks. A detailed analysis of the proposed approach on real-life Internet topologies with respect to network storage and traffic overhead, has provided a more realistic characterization. Thus, not only does the proposed approach lend well for simplified operations at scale but can also provide robust network-wide performance and security guarantees for Internet-wide attack attribution. Our final contribution here has introduced the notion of anonymity in the overall attack attribution process to significantly broaden its scope. The highly invasive nature of wide-spread data gathering for network traceback continues to violate one of the key principles of Internet use today - the ability to stay anonymous and operate freely without retribution. In this regard, we have successfully reconciled these mutually divergent requirements to make it not only economically feasible and politically viable but also socially acceptable. This work opens up several directions for future research - analysis of existing attack attribution techniques to identify further scope for improvements, incorporation of newer attributes into the design framework of the composable data model abstraction, and finally design of newer attack attribution techniques that comprehensively integrate the various attack prevention, mitigation and traceback techniques in an efficient manner

Combinatorial meshing for mechanical FEM

Diese Dissertation führt die Forschung zur Erzeugung von FEM Netzen für mechanische Simulationen fort. Zur zielgerichteten Steuerung der weiteren Forschung in diesem Feld wurde eine Umfrage zur Identifikation der Kerninteressen der Anwender durchgef¨uhrt. Das vorgestellte Verfahren des Combinatorial Meshing ist ein neuartiges Konzept im Bereich Grid Based Meshing. Im Gegensatz zu den kartesischen Gittern, die im Grid Based Meshing Anwendung finden wird ein an das Problem angepasstes Gitter genutzt. Dieses Precursor Mesh wird durch Analyse des CAD Strukturbaums der Geometrie gewählt. Die Zellen des Precursor Mesh werden mit vorberechneten Netzsegmenten – sogenannten Superelementen gefüllt. Die Wahl passender Superelemente wird als combinatorisches Optimierungsproblem modelliert. Dieses wird mit Hilfe von Answer Set Programming (ASP) und einem alternativen heuristischen Ansatz gelöst. Beide Verfahren werden in Hinblick auf Zeitkomplexität und Ergebnisqualität verglichen. Das resultierende Netz ist eine Grobe Näherung der Zielgeometrie, die an geometrische Elemente angebunden werden muss. Für diesen Prozess wird ein neuer Algorithmus vorgestellt, der automatisch identifizieren kann, an welche Geometrieelemente Oberflächenknoten des Netzes gebunden werden müssen um die Zielgeometrie möglichst exakt abzubilden. Für die Erzeugung der Superelemente wird ein neues Verfahren auf Basis von ASP entwickelt. Um die Generierung von FEM Netzen mit ASP zu ermöglichen, wird das Problem der Netzgenerierung als graphentheoretisches Problem modelliert. Dieses ist die Wahl eines optimalen Subgraphen aus einem Primärgraph. Dieses Problem wird mit einem ASP Solver für verschiedene Optimierungsziele gelöst. Die Graphenformulierung ist zudem ein Fortschritt im theoretischen Verständnis der Komplexität der Netzgenerierung.his dissertation advances the research of mesh generation for Finite Element Method simulation for mechanical applications. In order to target further research at user needs, a survey is conducted to identify the most pressing issues in FEM software. The concept of Combinatorial Meshing is proposed as a novel approach to grid based meshing. While conventional grid based meshing works on trivial Cartesian grids, the use of a Precursor Mesh instead of a grid is proposed. Appropriate Precursor Meshes are selected by analyzing the internal feature structure of the provided CAD data. The cells of this Precursor Mesh are then filled with precomputed mesh templates – called Super Elements. The selection of appropriate Super Elements is modeled as a combinatorial optimization problem. To solve this problem, Answer Set programming (ASP) and a heuristic approach are compared with respect to their time complexity and result quality. The resulting mesh is a rough approximation of the target geometry which then has to be fitted to the geometric entities. For this process a novel algorithm is presented which is able to automatically identify the geometric entities on which the surface nodes of the mesh have to be drawn in order to generate high quality meshes and correctly approximate the desired geometry. For the generation of Super Element Meshes, a novel approach based on ASP is developed. In order to enable meshing with ASP, a graph representation of a mesh is developed and the meshing process is formulated as a graph selection problem. It is then solved with an ASP solver for multiple optimization goals. The graph formulation will also aid the theoretical understanding of meshing complexity

3D object reconstruction using computer vision : reconstruction and characterization applications for external human anatomical structures

Tese de doutoramento. Engenharia Informática. Faculdade de Engenharia. Universidade do Porto. 201