Planarity Testing and Embedding
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
A Dynamic Data Structure for Planar Graph Embedding
October 1987 UILU-ENG-87-2265 ACT-83 COORDINATED SCIENCE LABORATORY College of Engineering Applied Computation Theory A DYNAMIC DATA STRUCTURE FOR PLANAR GRAPH EMBEDDING Roberto Tamassia UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Approved for Public Release. Distribution Unlimited. UNCLASSIFIED___________ SEÒjrtlfy CLASSIFICATION OP THIS PAGE REPORT DOCUMENTATION PAGE 1 a. REPORT SECURITY CLASSIFICATION 1b. RESTRICTIVE MARKINGS Unclassified None 2a. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILABIUTY OF REPORT 2b. DECLASSIFICATION / DOWNGRADING SCHEDULE Approved for public release; distribution unlimited 4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S) UILU-ENG-87-2265 ACT #83 6a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION Coordinated Science Lab (If applicati!a) University of Illinois N/A National Science Foundation 6c ADDRESS (City, Statt, and ZIP Coda) 7b. ADDRESS (City, Stata, and ZIP Coda) 1101 W. Springfield Avenue 1800 G Street, N.W. Urbana, IL 61801 Washington, D.C. 20550 8a. NAME OF FUNDING/SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (If applicatila) National Science Foundation ECS-84-10902 8c. ADDRESS (City, Stata, and ZIP Coda) 10. SOURCE OF FUNDING NUMBERS 1800 G Street, N.W. PROGRAM PROJECT TASK WORK UNIT Washington, D.C. 20550 ELEMENT NO. NO. NO. ACCESSION NO. A Dynamic Data Structure for Planar Graph Embedding 12. PERSONAL AUTHOR(S) Tamassia, Roberto 13a. TYPE OF REPORT 13b. TIME COVERED 14-^ A T E OP REPORT (Year, Month, Day) 5. PAGE COUNT Technical FROM______ TO 1987, October 26 ? 41 16. SUPPLEMENTARY NOTATION 17. COSATI CODES 18. SUBJECT TERMS (Continua on ravarsa if nacassary and identify by block number) FIELD GROUP SUB-GROUP planar graph, planar embedding, dynamic data structure, on-line algorithm, analysis of algorithms present a dynamic data structure that allows for incrementally constructing a planar embedding of a planar graph. -
1 Vertex Connectivity 2 Edge Connectivity 3 Biconnectivity
1 Vertex Connectivity So far we've talked about connectivity for undirected graphs and weak and strong connec- tivity for directed graphs. For undirected graphs, we're going to now somewhat generalize the concept of connectedness in terms of network robustness. Essentially, given a graph, we may want to answer the question of how many vertices or edges must be removed in order to disconnect the graph; i.e., break it up into multiple components. Formally, for a connected graph G, a set of vertices S ⊆ V (G) is a separating set if subgraph G − S has more than one component or is only a single vertex. The set S is also called a vertex separator or a vertex cut. The connectivity of G, κ(G), is the minimum size of any S ⊆ V (G) such that G − S is disconnected or has a single vertex; such an S would be called a minimum separator. We say that G is k-connected if κ(G) ≥ k. 2 Edge Connectivity We have similar concepts for edges. For a connected graph G, a set of edges F ⊆ E(G) is a disconnecting set if G − F has more than one component. If G − F has two components, F is also called an edge cut. The edge-connectivity if G, κ0(G), is the minimum size of any F ⊆ E(G) such that G − F is disconnected; such an F would be called a minimum cut.A bond is a minimal non-empty edge cut; note that a bond is not necessarily a minimum cut. -
Partial Duality and Closed 2-Cell Embeddings
Partial duality and closed 2-cell embeddings To Adrian Bondy on his 70th birthday M. N. Ellingham1;3 Department of Mathematics, 1326 Stevenson Center Vanderbilt University, Nashville, Tennessee 37240, U.S.A. [email protected] Xiaoya Zha2;3 Department of Mathematical Sciences, Box 34 Middle Tennessee State University Murfreesboro, Tennessee 37132, U.S.A. [email protected] April 28, 2016; to appear in Journal of Combinatorics Abstract In 2009 Chmutov introduced the idea of partial duality for embeddings of graphs in surfaces. We discuss some alternative descriptions of partial duality, which demonstrate the symmetry between vertices and faces. One is in terms of band decompositions, and the other is in terms of the gem (graph-encoded map) representation of an embedding. We then use these to investigate when a partial dual is a closed 2-cell embedding, in which every face is bounded by a cycle in the graph. We obtain a necessary and sufficient condition for a partial dual to be closed 2-cell, and also a sufficient condition for no partial dual to be closed 2-cell. 1 Introduction In this paper a surface Σ means a connected compact 2-manifold without boundary. By an open or closed disk in Σ we mean a subset of the surface homeomorphic to such a subset of R2. By a simple closed curve or circle in Σ we mean an image of a circle in R2 under a continuous injective map; a simple arc is a similar image of [0; 1]. The closure of a set S is denoted S, and the boundary is denoted @S. -
CLRS B.4 Graph Theory Definitions Unit 1: DFS Informally, a Graph
CLRS B.4 Graph Theory Definitions Unit 1: DFS informally, a graph consists of “vertices” joined together by “edges,” e.g.,: example graph G0: 1 ···················•······························· ····························· ····························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ············· ···· ···· ·············· 2•···· ···· ···· ··· •· 3 ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ······· ······· ······· ······· ···· ···· ···· ··· ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ··············· ···· ···· ··············· 4•························· ···· ···· ························· • 5 ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ····························· ····························· ···················•································ 6 formally a graph is a pair (V, E) where V is a finite set of elements, called vertices E is a finite set of pairs of vertices, called edges if H is a graph, we can denote its vertex & edge sets as V (H) & E(H) respectively if the pairs of E are unordered, the graph is undirected if the pairs of E are ordered the graph is directed, or a digraph two vertices joined by an edge -
Efficient Multicore Algorithms for Identifying Biconnected Components
International Journal of Networking and Computing { www.ijnc.org ISSN 2185-2839 (print) ISSN 2185-2847 (online) Volume 6, Number 1, pages 87{106, January 2016 Efficient Multicore Algorithms For Identifying Biconnected Components1 Meher Chaitanya [email protected] and Kishore Kothapalli [email protected] International Institute of Information Technology Hyderabad, Gachibowli, India 500 032. Received: July 30, 2015 Revised: October 26, 2015 Accepted: December 1, 2015 Communicated by Akihiro Fujiwara Abstract In this paper we design and implement an algorithm for finding the biconnected components of a given graph. Our algorithm is based on experimental evidence that finding the bridges of a graph is usually easier and faster in the parallel setting. We use this property to first decompose the graph into independent and maximal 2-edge-connected subgraphs. To identify the articulation points in these 2-edge connected subgraphs, we again convert this into a problem of finding the bridges on an auxiliary graph. It is interesting to note that during the conversion process, the size of the graph may increase. However, we show that this small increase in size and the run time is offset by the consideration that finding bridges is easier in a parallel setting. We implement our algorithm on an Intel i7 980X CPU running 12 threads. We show that our algorithm is on average 2.45x faster than the best known current algorithms implemented on the same platform. Finally, we extend our approach to dense graphs by applying the sparsification technique suggested by Cong and Bader in [7]. Keywords: Biconnected components, Least common ancestor, 2-edge connected components, Articulation points 1 Introduction The biconnected components of a given graph are its maximal 2-connected subgraphs. -
Bandwidth, Expansion, Treewidth, Separators and Universality for Bounded-Degree Graphs$
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 31 (2010) 1217–1227 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphsI Julia Böttcher a, Klaas P. Pruessmann b, Anusch Taraz a, Andreas Würfl a a Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany b Institute for Biomedical Engineering, University and ETH Zurich, Gloriastr. 35, 8092, Zürich, Switzerland article info a b s t r a c t Article history: We establish relations between the bandwidth and the treewidth Received 3 March 2009 of bounded degree graphs G, and relate these parameters to the size Accepted 13 October 2009 of a separator of G as well as the size of an expanding subgraph of Available online 24 October 2009 G. Our results imply that if one of these parameters is sublinear in the number of vertices of G then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each γ > 0 every n-vertex graph with minimum degree 3 C . 4 γ /n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large. ' 2009 Elsevier Ltd. -
[Cs.CG] 1 Nov 2019 Forbidden (See [11,24,36,38] for Surveys and Reports)
An Experimental Study of a 1-planarity Testing and Embedding Algorithm ? Carla Binucci, Walter Didimo, and Fabrizio Montecchiani Universit`adegli Studi di Perugia, Italy fcarla.binucci,walter.didimo,[email protected] Abstract. The definition of 1-planar graphs naturally extends graph planarity, namely a graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time, deciding whether a graph is 1-planar is NP-complete, even for restricted classes of graphs. Although several polynomial-time algorithms have been described for recognizing specific subfamilies of 1-planar graphs, no implementations of general algorithms are available to date. We investigate the feasibility of a 1-planarity test- ing and embedding algorithm based on a backtracking strategy. While the experiments show that our approach can be successfully applied to graphs with up to 30 vertices, they also suggest the need of more sophis- ticated techniques to attack larger graphs. Our contribution provides initial indications that may stimulate further research on the design of practical approaches for the 1-planarity testing problem. 1 Introduction The study of sparse nonplanar graphs is receiving increasing attention in the last years. One objective of this research stream is to extend the rich set of results about planar graphs to wider families of graphs that can better model real-world problems (see, e.g., [27,28,29,30,31]). Another objective is to create readable visualizations of nonplanar networks arising in various application scenarios (see, e.g., [23,39]). -
Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 16 (2015) An O(n) Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs Diptarka Chakraborty∗and Raghunath Tewari y Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, India April 27, 2015 Abstract Given a graph G and two vertices s and t in it, graph reachability is the problem of checking whether there exists a path from s to t in G. We show that reachability in directed layered planar graphs can be decided in polynomial time and O(n ) space, for any > 0. Thep previous best known space bound for this problem with polynomial time was approximately O( n) space [INP+13]. Deciding graph reachability in SC is an important open question in complexity theory and in this paper we make progress towards resolving this question. 1 Introduction Given a graph and two vertices s and t in it, the problem of determining whether there is a path from s to t in the graph is known as the graph reachability problem. Graph reachability problem is an important question in complexity theory. Particularly in the domain of space bounded computa- tions, the reachability problem in various classes of graphs characterize the complexity of different complexity classes. The reachability problem in directed and undirected graphs, is complete for the classes non-deterministic log-space (NL) and deterministic log-space (L) respectively [LP82, Rei08]. The latter follows due to a famous result by Reingold who showed that undirected reachability is in L [Rei08]. -
A Study of Solution Strategies for Some Graph
A STUDY OF SOLUTION STRATEGIES FOR SOME GRAPH EMBEDDING PROBLEMS A Synopsis Submitted in the partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics Submitted by Aditi Khandelwal Prof. Gur Saran Prof. Kamal Srivastava Supervisor Co-supervisor Department of Mathematics Department of Mathematics Prof. Ravinder Kumar Prof. G.S. Tyagi Head, Department of Head, Department of Physics Mathematics Dean, Faculty of Science Department of Mathematics Faculty of Science, Dayalbagh Educational Institute (Deemed University) Dayalbagh, Agra-282005 March, 2017. A STUDY OF SOLUTION STRATEGIES FOR SOME GRAPH EMBEDDING PROBLEMS 1. Introduction Many problems of practical interest can easily be represented in the form of graph theoretical optimization problems like the Travelling Salesman Problem, Time Table Scheduling Problem etc. Recently, the application of metaheuristics and development of algorithms for problem solving has gained particular importance in the field of Computer Science and specially Graph Theory. Although various problems are polynomial time solvable, there are large number of problems which are NP-hard. Such problems can be dealt with using metaheuristics. Metaheuristics are a successful alternative to classical ways of solving optimization problems, to provide satisfactory solutions to large and complex problems. Although they are alternative methods to address optimization problems, there is no theoretical guarantee on results [JJM] but usually provide near optimal solutions in practice. Using heuristic -
Strongly Connected Components and Biconnected Components
Strongly Connected Components and Biconnected Components Daniel Wisdom 27 January 2017 1 2 3 4 5 6 7 8 Figure 1: A directed graph. Credit: Crash Course Coding Companion. 1 Strongly Connected Components A strongly connected component (SCC) is a set of vertices where every vertex can reach every other vertex. In an undirected graph the SCCs are just the groups of connected vertices. We can find them using a simple DFS. This works because, in an undirected graph, u can reach v if and only if v can reach u. This is not true in a directed graph, which makes finding the SCCs harder. More on that later. 2 Kernel graph We can travel between any two nodes in the same strongly connected component. So what if we replaced each strongly connected component with a single node? This would give us what we call the kernel graph, which describes the edges between strongly connected components. Note that the kernel graph is a directed acyclic graph because any cycles would constitute an SCC, so the cycle would be compressed into a kernel. 3 1,2,5,6 4,7 8 Figure 2: Kernel graph of the above directed graph. 3 Tarjan's SCC Algorithm In a DFS traversal of the graph every SCC will be a subtree of the DFS tree. This subtree is not necessarily proper, meaning that it may be missing some branches which are their own SCCs. This means that if we can identify every node that is the root of its SCC in the DFS, we can enumerate all the SCCs. -
Confluent Hasse Diagrams
Confluent Hasse diagrams David Eppstein Joseph A. Simons Department of Computer Science, University of California, Irvine, USA. October 29, 2018 Abstract We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n) × O(n) grid in O(n2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) fea- tures in an O(n) × O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. 1 Introduction One of the most important aspects of a graph drawing is that it should be readable: it should convey the structure of the graph in a clear and concise way. Ease of understanding is difficult to quantify, so various proxies for readability have been proposed; one of the most prominent is the number of edge crossings. That is, we should minimize the number of edge crossings in our drawing (a planar drawing, if possible, is ideal), since crossings make drawings harder to read. Another measure of readability is the total amount of ink required by the drawing [1]. This measure can be formulated in terms of Tufte’s “data-ink ratio” [22,35], according to which a large proportion of the ink on any infographic should be devoted to information. Thus given two different ways to present information, we should choose the more succinct and crossing-free presentation. -
BOOK EMBEDDINGS of LAMAN GRAPHS 1. Definitions Definition 1
BOOK EMBEDDINGS OF LAMAN GRAPHS MICHELLE DELCOURT, MEGHAN GALIARDI, AND JORDAN TIRRELL Abstract. We explore various techniques to embed planar Laman graphs in books. These graphs can be embedded as pointed pseudo-triangulations and have many interesting properties. We conjecture that all planar Laman graphs can be embedded in two pages. 1. Definitions Definition 1. A graph G with n vertices and m edges is a Laman graph if m = 2n − 3 and every subgraph induced by k vertices (where k > 1) contains at most 2k − 3 edges. Definition 2. A book embedding of a graph G is a drawing with no crossings of G on a book; every edge is mapped to a page (a half-plane), and every vertex is mapped to the spine (the boundary shared by all the half-planes). Definition 3. The book thickness of a graph G, denoted bt(G) also known as page number, is the minimum number of pages needed to book embed G. 2. Finding Book Thickness Using Henneberg Constructions Motivated by Schnyder labelings for triangulations, Huemer and Kappes proposed an analogous binary labeling for quadrangulations which decom- poses the edge set in two directed trees rooted at a sink. A weak version of this labeling is valid for Laman graphs; however, the labeling does not guarantee a decomposition into two trees. The binary labeling of Laman graphs of Huemer and Kappes is based on Henneberg constructions. The Henneberg construction provides a decomposition into two trees whereas the labeling does not. A Henneberg construction consists of two moves: I. Adding a vertex of degree two II.