Simultaneous Embedding of Planar Graphs
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A Note on Simultaneous Embedding of Planar Graphs (Abstract)∗
EWCG 2005, Eindhoven, March 9–11, 2005 A Note on Simultaneous Embedding of Planar Graphs (Abstract)∗ Emilio Di Giacomo† Giuseppe Liotta† 1 Introduction 2 Preliminaries Let G1 and G2 be a pair of planar graphs such that The algorithms to compute a simultaneous embedding V (G1)=V (G2)=V .Asimultaneous embedding [6] of a pair of planar graphs and of a pair of trees pre- Ψ=(Γ1, Γ2)ofG1 and G2 is a pair of crossing-free sented in [6] are an elegant combination of the tech- drawings Γ1 and Γ2 of G1 and G2, respectively, such nique by Kaufmann and Wiese [7] for point-set em- ∈ that for every vertex v V we have Γ1(v)=Γ2(v). bedding and of the simultaneous embedding strategy ∈ ∩ If every edge e E(G1) E(G2) is represented with for two paths by Brass et al. [2]. In this section, we the same simple open Jordan curve both in Γ1 and in first recall the main ideas of Erten and Kobourov for Γ we say that Ψ is a simultaneous embedding with 2 simultaneous embedding of two planar graphs G1 and fixed edges. If the edges of G and G are repre- 1 2 G2 and then make a couple of observations on how to sented with straight-line segments in Γ1 and Γ2 we combine these ideas with known literature in order to say that Ψ is a simultaneous geometric embedding. extend some of the results in [6]. The existence of simultaneous geometric embeddings Suppose first that both G1 and G2 are Hamilto- for pairs of paths, cycles, and caterpillars is shown nian. -
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. -
Planar Graph Theory We Say That a Graph Is Planar If It Can Be Drawn in the Plane Without Edges Crossing
Planar Graph Theory We say that a graph is planar if it can be drawn in the plane without edges crossing. We use the term plane graph to refer to a planar depiction of a planar graph. e.g. K4 is a planar graph Q1: The following is also planar. Find a plane graph version of the graph. A B F E D C A Method that sometimes works for drawing the plane graph for a planar graph: 1. Find the largest cycle in the graph. 2. The remaining edges must be drawn inside/outside the cycle so that they do not cross each other. Q2: Using the method above, find a plane graph version of the graph below. A B C D E F G H non e.g. K3,3: K5 Here are three (plane graph) depictions of the same planar graph: J N M J K J N I M K K I N M I O O L O L L A face of a plane graph is a region enclosed by the edges of the graph. There is also an unbounded face, which is the outside of the graph. Q3: For each of the plane graphs we have drawn, find: V = # of vertices of the graph E = # of edges of the graph F = # of faces of the graph Q4: Do you have a conjecture for an equation relating V, E and F for any plane graph G? Q5: Can you name the 5 Platonic Solids (i.e. regular polyhedra)? (This is a geometry question.) Q6: Find the # of vertices, # of edges and # of faces for each Platonic Solid. -
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. -
Algorithms for Graph Visualization Flow Methods: Orthogonal Layout
Algorithms for Graph Visualization Flow Methods: Orthogonal Layout INSTITUT FUR¨ THEORETISCHE INFORMATIK · FAKULTAT¨ FUR¨ INFORMATIK Tamara Mchedlidze 18.01.2018 1 Dr. Tamara Mchedlidze · Algorithmen zur Visualisierung von Graphen Flussmethoden: orthogonales Graphenzeichnen Orthogonal Layout • Edges consist of vertical and horizontal segments • Applied in many areas ER diagram in OGDF UML diagram by Oracle Organigram of HS Limburg Circuit diagram by Jeff Atwood 2 - 1 Dr. Tamara Mchedlidze · Algorithmen zur Visualisierung von Graphen Flussmethoden: orthogonales Graphenzeichnen Orthogonal Layout • Edges consist of vertical and horizontal segments • Applied in many areas Aesthetic functions? ER diagram in OGDF UML diagram by Oracle Organigram of HS Limburg Circuit diagram by Jeff Atwood 2 - 2 Dr. Tamara Mchedlidze · Algorithmen zur Visualisierung von Graphen Flussmethoden: orthogonales Graphenzeichnen Orthogonal Layout • Edges consist of vertical and horizontal segments • Applied in many areas Aesthetic functions: • number of bends • length of edges • ER diagram in OGDF width, height, area • monotonicity of edges UML diagram by Oracle • ... Organigram of HS Limburg Circuit diagram by Jeff Atwood 2 - 3 Dr. Tamara Mchedlidze · Algorithmen zur Visualisierung von Graphen Flussmethoden: orthogonales Graphenzeichnen (Planar) Orthogonal Drawings Three-step approach: Topology { Shape { Metrics [Tamassia SIAM J. Comput. 1987] V = fv1; v2; v3; v4g E = fv1v2; v1v3; v1v4; v2v3; v2v4g 3 - 1 Dr. Tamara Mchedlidze · Algorithmen zur Visualisierung von Graphen Flussmethoden: orthogonales Graphenzeichnen (Planar) Orthogonal Drawings Three-step approach: Topology { Shape { Metrics [Tamassia SIAM J. Comput. 1987] V = fv1; v2; v3; v4g E = fv1v2; v1v3; v1v4; v2v3; v2v4g combinatorial embedding/ planarization 4 3 Reduce Crossings 2 1 3 - 2 Dr. Tamara Mchedlidze · Algorithmen zur Visualisierung von Graphen Flussmethoden: orthogonales Graphenzeichnen (Planar) Orthogonal Drawings Three-step approach: Topology { Shape { Metrics [Tamassia SIAM J. -
Graph Drawing Dagstuhl Seminar
05191 Abstracts Collection Graph Drawing Dagstuhl Seminar Michael Jünger1, Stephen Kobourov2 and Petra Mutzel3 1 Univ. of Köln, DE 2 Univ. of Arizona, US [email protected] 3 Univ. of Dortmund, DE [email protected] Abstract. From 08.05. to 13.05.05, the Dagstuhl Seminar 05191 Graph Drawing was held in the International Conference and Research Cen- ter (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. Keywords. Graph Drawing, Visualization, Layout Algorithms, Inter- active Visualization 05191 Executive Summary Graph Drawing This paper summarizes the topics, aims, and achievements of the Dagstuhl Sem- inar 05191 on Graph Drawing. Keywords: Graph drawing Joint work of: Jünger, Michael; Kobourov, Stephen; Mutzel, Petra Full Paper: http://drops.dagstuhl.de/opus/volltexte/2006/342 05191 Open Problem Session Report This is a report on an informal session intended to stimulate communication and sharing of problems. Hence the attributions and citations may contain inaccu- racies, and are certainly not complete. Keywords: Open problems, graph drawing Joint work of: Whitesides, Sue Full Paper: http://drops.dagstuhl.de/opus/volltexte/2006/338 Dagstuhl Seminar Proceedings 05191 Graph Drawing http://drops.dagstuhl.de/opus/volltexte/2006/348 2 M. Jünger, S. Kobourov and P. Mutzel Matrix Zoom: A Visual Interface to Semi-external Graphs James Abello (Rutgers Univ. -
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 -
Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges
Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges J. Joseph Fowler1, Michael J¨unger2, Stephen Kobourov1, and Michael Schulz2 1 University of Arizona, USA {jfowler,kobourov}@cs.arizona.edu ⋆ 2 University of Cologne, Germany {mjuenger,schulz}@informatik.uni-koeln.de ⋆⋆ Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for when a pair of graphs whose union is homeomorphic to K5 or K3,3 can have an SEFE. This allows us to determine which (outer)planar graphs always an SEFE with any other (outer)planar graphs. In both cases, we provide efficient al- gorithms to compute the simultaneous drawings. Finally, we provide an linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has an SEFE. 1 Introduction In many practical applications including the visualization of large graphs and very-large-scale integration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex set can be used with multiple edge sets that each correspond to different edge colors or circuit layers. While the pairwise union of all edge sets may be nonplanar, a planar drawing of each layer may be possible, as crossings between edges of distinct edge sets are permitted. -
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. -
Separator Theorems and Turán-Type Results for Planar Intersection Graphs
SEPARATOR THEOREMS AND TURAN-TYPE¶ RESULTS FOR PLANAR INTERSECTION GRAPHS JACOB FOX AND JANOS PACH Abstract. We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs. For instance, we show that any collection C of Jordan curves in the plane with a total of m p 2 crossings has a partition into three parts C = S [ C1 [ C2 such that jSj = O( m); maxfjC1j; jC2jg · 3 jCj; and no element of C1 has a point in common with any element of C2. These results are used to obtain various properties of intersection patterns of geometric objects in the plane. In particular, we prove that if a graph G can be obtained as the intersection graph of n convex sets in the plane and it contains no complete bipartite graph Kt;t as a subgraph, then the number of edges of G cannot exceed ctn, for a suitable constant ct. 1. Introduction Given a collection C = fγ1; : : : ; γng of compact simply connected sets in the plane, their intersection graph G = G(C) is a graph on the vertex set C, where γi and γj (i 6= j) are connected by an edge if and only if γi \ γj 6= ;. For any graph H, a graph G is called H-free if it does not have a subgraph isomorphic to H. Pach and Sharir [13] started investigating the maximum number of edges an H-free intersection graph G(C) on n vertices can have. If H is not bipartite, then the assumption that G is an intersection graph of compact convex sets in the plane does not signi¯cantly e®ect the answer. -
Open Problems on Graph Drawing
Open problems on Graph Drawing Alexandros Angelopoulos Corelab E.C.E - N.T.U.A. February 17, 2014 Outline Introduction - Motivation - Discussion Variants of thicknesses Thickness Geometric thickness Book thickness Bounds Complexity Related problems & future work 2/ Motivation: Air Traffic Management Separation -VerticalVertical - Lateral - Longitudinal 3/ Motivation: Air Traffic Management : Maximization of \free flight” airspace c d c d X f f i1 i4 i0 i2 i3 i5 e e Y a b a b 8 Direct-to flight (as a choice among \free flight") increases the complexity of air traffic patterns Actually... 4 Direct-to flight increases the complexity of air traffic patterns and we have something to study... 4/ Motivation: Air Traffic Management 5/ How to model? { Graph drawing & thicknesses Geometric thickness (θ¯) Book thickness (bt) Dillencourt et al. (2000) Bernhart and Kainen (1979) : only straight lines : convex positioning of nodes v4 v5 v1 v2 θ(G) ≤ θ¯(G) ≤ bt(G) v5 v3 v0 v4 4 v1 Applications in VLSI & graph visualizationv3 Thickness (θ) v0 8 θ, θ¯, bt characterize the graph (minimizations over all allowedv2 drawings) Tutte (1963), \classical" planar decomposition 6/ Geometric graphs and graph drawings Definition 1.1 (Geometric graph, Bose et al. (2006), many Erd¨ospapers). A geometric graph G is a pair (V (G);E(G)) where V (G) is a set of points in the plane in general position and E(G) is set of closed segments with endpoints in V (G). Elements of V (G) are vertices and elements of E(G) are edges, so we can associate this straight-line drawing with the underlying abstract graph G(V; E). -
Ma/CS 6B Class 8: Planar and Plane Graphs
1/26/2017 Ma/CS 6b Class 8: Planar and Plane Graphs By Adam Sheffer The Utilities Problem Problem. There are three houses on a plane. Each needs to be connected to the gas, water, and electricity companies. ◦ Is there a way to make all nine connections without any of the lines crossing each other? ◦ Using a third dimension or sending connections through a company or house is not allowed. 1 1/26/2017 Rephrasing as a Graph How can we rephrase the utilities problem as a graph problem? ◦ Can we draw 퐾3,3 without intersecting edges. Closed Curves A simple closed curve (or a Jordan curve) is a curve that does not cross itself and separates the plane into two regions: the “inside” and the “outside”. 2 1/26/2017 Drawing 퐾3,3 with no Crossings We try to draw 퐾3,3 with no crossings ◦ 퐾3,3 contains a cycle of length six, and it must be drawn as a simple closed curve 퐶. ◦ Each of the remaining three edges is either fully on the inside or fully on the outside of 퐶. 퐾3,3 퐶 No 퐾3,3 Drawing Exists We can only add one red-blue edge inside of 퐶 without crossings. Similarly, we can only add one red-blue edge outside of 퐶 without crossings. Since we need to add three edges, it is impossible to draw 퐾3,3 with no crossings. 퐶 3 1/26/2017 Drawing 퐾4 with no Crossings Can we draw 퐾4 with no crossings? ◦ Yes! Drawing 퐾5 with no Crossings Can we draw 퐾5 with no crossings? ◦ 퐾5 contains a cycle of length five, and it must be drawn as a simple closed curve 퐶.