Academia.edu no longer supports Internet Explorer.
To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser.
People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Algorithmica, 2007
We consider graphs that can be embedded on a surface of bounded genus such that each edge has a bounded number of crossings. We prove that many optimization problems, including maximum independent set, minimum vertex cover, minimum dominating set and many others, admit polynomial time approximation schemes when restricted to such graphs. This extends previous results by Baker and Eppstein [3] to a much broader class of graphs.
Lecture Notes in Computer Science, 2015
DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Algorithmic graph theory is a classical area of research by now and has been rapidly expanding during the last three decades. In many different contexts of computer science and applications, modelling problems by graphs is a natural and canonical process. Graph-theoretic concepts and algorithms play an important role in many fields of application, e.g. in communication network design, VLSI-design, CAD, traffic optimization or network visualization.
Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(g n log n) time, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [Multiple-source shortest paths in planar graphs. In Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of f . As an application of our algorithm, we describe algorithms to compute a shortest non-contractible or non-separating cycle in embedded, undirected graphs in O(g 2 n log n) time.
Open Computer Science, 2015
We propose several new heuristics for the twopage book crossing problem, which are based on recent algorithms for the corresponding one-page problem. Especially, the neural network model for edge allocation is combined for the first time with various one-page algorithms. We investigate the performance of the new heuristics by testing them on various benchmark test suites. It is found out that the new heuristics outperform the previously known heuristics and produce good approximations of the planar crossing number for severalwell-known graph families. We conjecture that the optimal two-page drawing of a graph represents the planar drawing of the graph.
Figure 1.3.1: Organization of our book consisting of nine chapters. The directed acyclic graph illustrates a possible teaching strategy.
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with O(log 2 n)• (n + OPT) crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Planarization problem on graph G, then we can efficiently find a drawing of G with at most poly(d) • k • (k + OPT) crossings, where d is the maximum degree in G. This result implies an O(n • poly(d) • log 3/2 n)approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.
The simplest graph drawing method is that of putting the vertices of a graph on a line (spine) and drawing the edges as half-circles on k half planes (pages). Such drawings are called kpage book drawings and the minimal number of edge crossings in such a drawing is called the k-page crossing number. In a one-page book drawing, all edges are placed on one side of the spine, and in a two-page book drawing all edges are placed either above or below the spine. The one-page and two-page crossing numbers of a graph provide upper bounds for the standard planar crossing. In this paper, we derive the exact one-page crossing numbers for four-row meshes, present a new proof for the one-page crossing numbers of Halin graphs, and derive the exact two-page crossing numbers for circulant graphs Cn(1, n 2). We also give explicit constructions of the optimal drawings for each kind of graphs.
Lecture Notes in Computer Science, 1999
The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V 0 ; V 1 ; E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V 0 and V 1 can be permuted arbitrarily | both this and the case where the order of one vertex set is xed are NP-hard. Two new heuristics that perform well on sparse graphs such as occur in circuit layout problems are presented. The new heuristics outperform existing heuristics on graph classes that range from applicationspeci c to random. Our experimental design methodology ensures that di erences in performance are statistically signi cant and not the result of minor variations in graph structure or input order.
2018
We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For c=1 there are only two such graphs without degree-2 vertices, K_5 and K_3,3, but for any fixed c>1 there exist infinitely many c-crossing-critical graphs. It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every c-crossing-critical graph can be obtained from a c-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the c-crossing-critical ...
Russian Mathematical Surveys, 2003
Combinatorica, 1997
We show that if a graph of v vertices can be drawn in the plane so that every edge crosses at most k>0 others, then its number of edges cannot exceed 4.108V"kv. For k<4, we establish a better bound, (kq-3)(v-2), which is tight for k-= 1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges.
1995
LINK is a set of C++ class libraries that supports applications in discrete mathematics. The libraries include a commandline interpreter and a graphical user interface that allow access to basic data structures such as Sets and Lists, and a graph hierarchy that includes undirected, directed, and \mixed" hypergraphs and graphs. A \mixed" graph may contain both directed and undirected edges. Many standard data structures including arrays, lists, heaps and binary search trees are within a Container hierarchy. Sets and Sequences are supported within a Collection hierarchy. The data structure hierarchies enable the user to experiment with competing data structure implementations, and with more complex and sophisticated data structures. If an algorithm has several possible choices of a data structure to be used, a single object can be created that is templated with the particular data structure desired. LINK also contains a set of graph generators, layout algorithms for hypergraphs and binary graphs, and numerous graph algorithms.
Springer eBooks, 2022
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The field of mathematics plays vital role in various fields. One of the important areas in mathematics is graph theory which is used in structural models. This structural arrangements of various objects or technologies lead to new inventions and modifications in the existing environment for enhancement in those fields. This Paper describes the description of graph theory.
We are currently preparing students for jobs that don't yet exist using technologies that haven't been invented in order to solve problems we don't even know are problems yet." ~ Karl Fisch
… : Proceedings of the …
Admissible Graph Rewriting and Narrowing Rachid Echahed and Jean-Christophe Janodet Laboratoire LEIBNIZ, IMAG, CNRS 46, avenue Felix Viallet F-38031 Grenoble-France Rachid. Echahed@ imag. fr; Jean-Christophe. Janodet@ imag. fr Abstract We address ...
2019
Networks are used as models of complex systems in a wide variety of subjects, including chemistry, biology, sociology, engineering and computer science. We develop an interactive tool that contains more than a billion networks and related parameters. It is designed to help researchers gain insights into the structure of networks and develop new theorems about them. We also introduce and analyse the problem of finding the most commonly occurring substructure inside a network.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.