DOCSLIB.ORG

Algorithms for Testing and Embedding Planar Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algorithms for Testing and Embedding Planar Graphs 1 Algorithms for Testing and Embedding Planar Graphs Zhigang Jiang, University of Windsor A planar graph is a graph which can be drawn in the plane without any edges crossing. Some pictures of a planar graph might have crossing edges, but it's possible to redraw the picture to eliminate the crossings. Planar graphs and their relatives, appear in many practical applications. For example, in the design of complex radioelectronic circuits using printed circuit board, one problem is to arrange the elements so that the conductors connecting them do not intersect each other. This survey contains two crucial parts: algorithms for embedding graphs into planarity which is about embedding a given graph G into planarities , and planarity testing algorithms which is about testing whether a given graph G can be embedded into planarity or not. Categories and Subject Descriptors: F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical Algorithms and Problems—Geometrical problem and computations; I.3.5 [Computer Graphics]: Computational Geometry and Object Model- ing—Geometric algorithms,languages, and systems General Terms: Algorithms Additional Key Words and Phrases: planar graph, embedding graphs into planarity algorithms, planarity testing algorithms Contents 1 Introduction 2 2 Embedding graphs into planarity 3 2.1 embedding algorithms donot use PQ-trees . .3 2.1.1 A planarity embedding algorithm based on the Kuratowski theorem . .3 2.1.2 An embedding algorithm based on open ear decomposition . .3 2.1.3 A simplified o (n) planar embedding algorithm for biconnected graphs . .4 2.1.4 Lempel,Even,and Cederbaum planarity method . .5 2.2 Embedding algorithms using PQ-trees . .5 2.2.1 Embedding planar graphs using PQ-tree algorithms . .6 2.2.2 An algorithm is on the basis of vertex addition for planarity testing and uses PQ- trees to embed graphs into planarites . .6 2.2.3 An algorithm modifies PQ-trees to embed graphs into planarities . .7 3 Planarity testing algorithms 10 3.1 Planarity testing algorithm based on 3-connectivity . 10 3.1.1 A new planarity test based on 3-connectivity . 10 3.2 Planarity testing algorithms based on DFS . 11 3.2.1 An algorithm uses DFS to test the planarity of a graph . 11 3.2.2 A simple planarity testing algorithm based on DFS search tree and modified Ver- tex addition approach . 12 4 Concluding Comments 13 5 Annotation 16 5.1 Bruno 1970 . 16 5.2 Boyer 1999 . 17 , Vol. V, No. N, Article 1, Publication date: January 20YY. 1:2 • Zhigang Jiang 5.3 Boyer 2004 . 17 5.4 Chiba 1984 . 18 5.5 Hopcroft 1974 . 19 5.6 Haeupler 2008 . 20 5.7 Mei 1970 . 21 5.8 Ozawa 1991 . 21 5.9 Ramachandran 1989 . 22 5.10 Shih 1993 . 23 1. INTRODUCTION A graph is planar if it can be drawn in the plane without any crossing edges. That is, each vertex is located at one point of the plane, and a curve from one point to another is drawn between the points corresponding to vertices connected by an edge. A practical reason for studying planar graphs is that they, and their relatives, appear in many practical applications. The study of two-dimensional images often results in problems related to planar graphs, as does the solution of many problems on the two- dimensional surface of our earth, for example, In Gas, Water, Electricity Problem, Is there any way to connect each of three houses to each of the three utilities(Gas, water, and electricity) in such a way that none of the supply lines cross? This survey is about that embedding a given graph into planarity using various embedding algo- rithms, or testing whether a given graph can be embedded into planarity or not using different testing algorithms. The papers in the survey have been found in Google Scholar, ACM, IEEE, and Journal of Computer and System Sciences and others. Among these papers that I found, there are 11 journal papers:[Bruno et al. 1970], [Hopcroft and Tarjan 1974], [Booth and Lueker 1976], [Williamson 1980], [Chiba et al. 1984], [Ramachandran and Reif 1994], [Juvan et al. 1997], [Jianer and Chen 1997], [Wei-Kuan and Wen-Lian 1999], [Boyer et al. 2004], [Haeupler and Tarjan 2008]; there are 7 conference papers: [Mei and Gibbs 1970], [Ramachan- dran and Reif 1989], [Ozawa 1991], [Shih and Hsu 1993], [Boyer and Myrvold 1999], [Kawarabayashi et al. 2008], [Kawarabayashi 2009]; there is one unpublished paper: [Boyer et al. 2003], and there is one chapter of a book: [Perunicic and Duric 1985]. To study the planar graphs problem, there are two important parts that need to be considered: embedding graphs into planarity and planarity testing. Therefore, in the rest of the survey, there are two sections: embedding graphs into planarity algorithms and planarity testing algorithms. The section on algorithms for embedding graphs into planarity, it includes embedding algorithms based on PQ-tree and other embedding algorithms, and in the other section, includes planarity algorithm based on DFS and planarity algorithm based on 3-connectivity. The relationships of these papers are on follow: Hopcroft and Tarjan [1974] is cited by: Chiba et al. [1984], Ramachandran and Reif [1989], Shih and Hsu [1993],Boyer and Myrvold [1999] and Boyer et al. [2004]; Booth and Lueker [1976] is cited by: Chiba et al. [1984], Ramachandran and Reif [1989], Boyer and Myrvold [1999], Boyer et al. [2004], and ?]; Williamson [1980] is cited by: Boyer and Myr- vold [1999]; Shih and Hsu [1993] is cited by: Boyer et al. [2004], and Haeupler and Tarjan [2008]; Ramachandran and Reif [1994] is cited by: Boyer et al. [2004]; Boyer and Myrvold [1999] is cited by: Haeupler and Tarjan [2008]; Wei-Kuan and Wen-Lian [1999] is cited by: Boyer et al. [2004], Haeupler and Tarjan [2008]; and Boyer et al. [2004] is cited by Haeupler and Tarjan [2008]. The relationships indicate that [Hopcroft and Tarjan 1974] and [Booth and Lueker 1976] are two very important papers in this area. Many of embedding algorithms use PQ-trees which are very important data structures for embed- ding algorithms and these papers from year to year finds a way to improve the previous algorithm. , Vol. V, No. N, Article 1, Publication date: January 20YY. Algorithms for Tesing and Embedding Planar Graphs • 1:3 2. EMBEDDING GRAPHS INTO PLANARITY The problem addressed by the authors is how to embed a given graph G into planarity. Firstly, testing whether the graph can be embedded into planarity or not, secondly, if the graph can be embedded into planarity, draw the planar graph of the given graph, and This section consists of two subsections: Embedding planar graphs using PQ-trees, and other embedding algorithms. 2.1 embedding algorithms donot use PQ-trees In this subsection, there are some algorithms to embed a given grap into planarity, but they do not use PQ-tree data structure, there are 4 papers in this subsection: A planarity algorithm based on the kuratowski theorem, An optimal parallel algorithm for graph planarity, Stop minding your p’s and q’s: A simplified O(n) planar embedding algorithm,and Lempel, even and cederbaum planarity method. 2.1.1 A planarity embedding algorithm based on the Kuratowski theorem. Mei and Gibbs [1970] do not refer to any previous work. Mei and Gibbs [1970] states that all of the previous work are not dicrect applications of Kuratowski theorem. In this paper, Mei and Gibbs [1970] present an algorithm which is a direct application of The Kura- towski Theorem, and based on the observation that a Kuratowski graph can be spanned by the union of two of its circuits. The details of the algorithm are as follows Mei and Gibbs [1970, p.93]: (1) Generate all the circuits of length five or greater, for a given graph G. (2) For any two circuits C1 and C2, let SG(C1) = <V1,C1>and SG(C2) = <V2,C2>. If V1 \ V2 has exactly five elements and C1 \ C2 = ;, go to next step, otherwise, go to step 4. (3) Let (vi1,vi2,vi3,vi4, vi5) be the five elements of V1 \ V2 ordered in this cyclic order and Trace SG(C1) in one direction, trace SG(C2) and check to see whether all of the five elements can be place in a ∗ cyclic order (vi1,vi2,vi3,vi4, vi5) or not, if it can, SG(C1 [ C2) is a K5 graph and graph G is non-planar, otherwise, go to last step. (4) If V1 \ V2 has more than five elements and C1 \ C2 has more than two elements, go to the step 5, otherwise go to the last step. (5) Form the vertex adjacency matrix M = (mij) of SG(C1 [ C2), if vi,vj62C1 [C2, mij = 0; if vi,vj2C1-C2, mij = 1; if vi,vj2C2-C1, mij = 2; if vi,vj2C1 \C2, mij = 3. (6) Check the entries of the matrix row b row once, and result in a cubic graph G = <V, E> with six vertices. (7) If there are no more pairs of circuits to be considered the graph G is planar, otherwise go to step 2 and choose another pairs of circuits C1 and C2 of G, and go to step 2. Mei and Gibbs [1970] do not mention any specific implementation or experiments. Mei and Gibbs [1970] claim that the algorithm could be very efficient for a small number of graphs or graphs that do not have a large number of circuits, but it is not efficient for a large graph,and the time complexity of this algorithm is O(n). Mei and Gibbs [1970] claim that the first step (the generation of circuits) of the algorithm can be executed first and the generated circuits can be stored on some form of auxiliary storage.
Load more

Recommended publications
  • [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]).
    [Show full text]
  • Graph Planarity
    Graph Planarity Alina Shaikhet Outline ▪ Definition. ▪ Motivation. ▪ Euler’s formula. ▪ Kuratowski’s theorems. ▪ Wagner’s theorem. ▪ Planarity algorithms. ▪ Properties. ▪ Crossing Number Definitions ▪ A graph is called planar if it can be drawn in a plane without any two edges intersecting. ▪ Such a drawing we call a planar embedding of the graph. ▪ A plane graph is a particular planar embedding of a planar graph. Motivation ▪ Circuit boards. Motivation ▪ Circuit boards. ▪ Connecting utilities (electricity, water, gas) to houses. Motivation ▪ Circuit boards. ▪ Connecting utilities (electricity, water, gas) to houses. Motivation ▪ Circuit boards. ▪ Connecting utilities (electricity, water, gas) to houses. ▪ Highway / Railroads / Subway design. self loops multi-edges Euler’s formula. Consider any plane embedding of a planar connected graph. Let V - be the number of vertices, E - be the number of edges and F - be the number of faces (including the single unbounded face), Then 푉 − 퐸 + 퐹 = 2. Euler formula gives the necessary condition for a graph to be planar. self loops multi-edges Euler’s formula. Consider any plane embedding of a planar connected graph. Let V - be the number of vertices, E - be the number of edges and F - be the number of faces (including the single unbounded face), Then 푉 − 퐸 + 퐹 = 2. Then 푉 − 퐸 + 퐹 = 퐶 + 1. C - is the number of connected components. 푉 − 퐸 + 퐹 = 2 Euler’s formula. 푉 = 6 퐸 = 12 퐹 = 8 푉 − 퐸 + 퐹 = 2 6 − 12 + 8 = 2 푉 − 퐸 + 퐹 = 2 Corollary 1 Let G be any plane embedding of a connected planar graph with 푉 ≥ 3 vertices. Then 1.
    [Show full text]
  • Planarity Testing Revisited
    Electronic Colloquium on Computational Complexity, Report No. 9 (2011) Planarity Testing Revisited Samir Datta, Gautam Prakriya Chennai Mathematical Institute India fsdatta,[email protected] Abstract. Planarity Testing is the problem of determining whether a given graph is planar while planar embedding is the corresponding construction problem. The bounded space complexity of these problems has been determined to be exactly Logspace by Allender and Mahajan [AM00] with the aid of Reingold's result [Rei08]. Unfortunately, the algorithm is quite daunting and generalizing it to say, the bounded genus case seems a tall order. In this work, we present a simple planar embedding algorithm running in logspace. We hope this algorithm will be more amenable to generalization. The algorithm is based on the fact that 3-connected planar graphs have a unique embedding, a variant of Tutte's criterion on conflict graphs of cycles and an explicit change of cycle basis. We also present a logspace algorithm to find obstacles to planarity, viz. a Kuratowski minor, if the graph is non-planar. To the best of our knowledge this is the first logspace algorithm for this problem. 1 Introduction Planarity Testing, the problem of determining whether a given graph is planar (i.e. the vertices and edges can be drawn on a plane with no edge intersections except at their end- points) is a fundamental problem in algorithmic graph theory. Along with the problem of actually obtaining a planar embedding, it is a prerequisite for many an algorithm designed to work specifically for planar graphs. Our focus is on the bounded space complexity of the planarity embedding problem be- cause we know that many graph theoretic problems like reachability [BTV09], perfect match- ing [DKR10,DKT10], and even isomorphism [DLN08,DLN+09] have efficient bounded space algorithms when provided graphs embedded on the plane.
    [Show full text]
  • Graph Planarity Testing with Hierarchical Embedding Constraints
    Graph Planarity Testing with Hierarchical Embedding Constraints Giuseppe Liottaa, Ignaz Rutterb, Alessandra Tappinia,∗ aDipartimento di Ingegneria, Universit`adegli Studi di Perugia, Italy bDepartment of Computer Science and Mathematics, University of Passau, Germany Abstract Hierarchical embedding constraints define a set of allowed cyclic orders for the edges incident to the vertices of a graph. These constraints are expressed in terms of FPQ-trees. FPQ-trees are a variant of PQ-trees that includes F-nodes in addition to P- and to Q-nodes. An F-node represents a permutation that is fixed, i.e., it cannot be reversed. Let G be a graph such that every vertex of G is equipped with a set of FPQ-trees encoding hierarchical embedding constraints for its incident edges. We study the problem of testing whether G admits a planar embedding such that, for each vertex v of G, the cyclic order of the edges incident to v is described by at least one of the FPQ-trees associated with v. We prove that the problem is fixed-parameter tractable for biconnected graphs, where the parameters are the treewidth of G and the number of FPQ- trees associated with every vertex of G. We also show that the problem is NP-complete if parameterized by the number of FPQ-trees only, and W[1]-hard if parameterized by the treewidth only. Besides being interesting on its own right, the study of planarity testing with hierarchical embedding constraints can be used to address other planarity testing problems. In particular, we apply our techniques to the study of NodeTrix planarity testing of clustered graphs.
    [Show full text]
  • Planar Graphs
    PLANAR GRAPHS Basic definitions Isomorphic graphs Two graphs G1(V1,E1) and G2(V2,E2) are isomorphic if there is a one-to-one correspondence F of their vertices such that the following holds: - " u,v ÎV1 , uv ÎE1, => F(u)F(v) Î E2 - " x,y ÎV1, xy ÏE1 => F(x)F(y) Ï E2 Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph Planar graph A graph is called Planar, if it is isomorphic with a Plane graph Phases A planar representation of a graph divides the plane in to a number of connected regions, called faces, each bounded by edges of the graph. For every graph G, we denote n(G) the number of vertices , e(G) the number of edges, f(G) the number of faces. Degree We define the degree of a face d(f), to be the number of edges bounding the face f. Examples The following graphs are isomorphic to 4 (the complete graph with 4 vertices) f2 f1 G2 G1 f3 f4 f1' f4' f2' f3' G3 Figure 1: 3 isomorphic graphs G2 and G3 are plane graphs, G1 is not. G1 is planar, as it is isomorphic to G2 and G3. G2 and G3 are called planar embeddings of K4 f1, f2 and f3 (exterior phase) are phases of G2, with degree d(f1)=d(f2)=d(f3)=3. Theorem 1 A graph is embeddable in the sphere if and only if it is embeddable in the plane. Proof.
    [Show full text]
  • Planarity Testing and Embedding
    1 Planarity Testing and Embedding 1.1 Introduction................................................. 1 1.2 Properties and Characterizations of Planar Graphs . 2 Basic Definitions • Properties • Characterizations 1.3 Planarity Problems ........................................ 7 Constrained Planarity • Deletion and Partition Problems • Upward Planarity • Outerplanarity 1.4 History of Planarity Algorithms.......................... 10 1.5 Common Algorithmic Techniques and Tools ........... 10 1.6 Cycle-Based Algorithms ................................... 11 Adding Segments: The Auslander-Parter Algorithm • Adding Paths: The Hopcroft-Tarjan Algorithm • Adding Edges: The de Fraysseix-Ossona de Mendez-Rosenstiehl Algorithm 1.7 Vertex Addition Algorithms .............................. 17 The Lempel-Even-Cederbaum Algorithm • The Shih-Hsu Algorithm • The Boyer-Myrvold Algorithm 1.8 Frontiers in Planarity ...................................... 31 Simultaneous Planarity • Clustered Planarity • Maurizio Patrignani Decomposition-Based Planarity Roma Tre University References ................................................... ....... 34 1.1 Introduction Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing algorithmic problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs has been known since 1930, the first linear-time solution to this problem was found only in the 1970s. Planar graphs
    [Show full text]
  • An Annotated Bibliography on 1-Planarity
    An annotated bibliography on 1-planarity Stephen G. Kobourov1, Giuseppe Liotta2, Fabrizio Montecchiani2 1Dep. of Computer Science, University of Arizona, Tucson, USA [email protected] 2Dip. di Ingegneria, Universit`adegli Studi di Perugia, Perugia, Italy fgiuseppe.liotta,[email protected] July 21, 2017 Abstract The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs. 1 Introduction Relational data sets, containing a set of objects and relations between them, are commonly modeled by graphs, with the objects as the vertices and the relations as the edges. A great deal is known about the structure and properties of special types of graphs, in particular planar graphs. The class of planar graphs is fundamental for both Graph Theory and Graph Algorithms, and is extensively studied. Many structural properties of planar graphs are known and these properties can be used in the development of efficient algorithms for planar graphs, even where the more general problem is NP-hard [11].
    [Show full text]
  • Strip Planarity Testing of Embedded Planar Graphs
    Strip Planarity Testing of Embedded Planar Graphs Patrizio Angelini1, Giordano Da Lozzo1, Giuseppe Di Battista1, Fabrizio Frati2 1 Dipartimento di Ingegneria, Roma Tre University, Italy {angelini,dalozzo,gdb}@dia.uniroma3.it 2 School of Information Technologies, The University of Sydney, Australia [email protected] Abstract. In this paper we introduce and study the strip planarity testing prob- lem, which takes as an input a planar graph G(V,E) and a function γ : V → {1, 2,...,k} and asks whether a planar drawing of G exists such that each edge is monotone in the y-direction and, for any u, v ∈ V with γ(u) <γ(v), it holds y(u) <y(v). The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, up- ward planarity, and level planarity. We show that the problem is polynomial-time solvable if G has a fixed planar embedding. 1 Introduction Testing the planarity of a given graph is one of the oldest and most deeply investigated problems in algorithmic graph theory. A celebrated result of Hopcroft and Tarjan [20] states that the planarity testing problem is solvable in linear time. A number of interesting variants of the planarity testing problem have been con- sidered in the literature [25]. Such variants mainly focus on testing, for a given planar graph G, the existence of a planar drawing of G satisfying certain constraints. For ex- ample the partial embedding planarity problem [1,22] asks whether a plane drawing G of a given planar graph G exists in which the drawing of a subgraph H of G in G coincides with a given drawing H of H.
    [Show full text]
  • The Complexity of Planarity Testing∗
    The complexity of planarity testing∗ Eric Allender† Dept. of Computer Science, Rutgers University, Piscataway, NJ, USA. Email: [email protected] Meena Mahajan‡ The Institute of Mathematical Sciences, Chennai 600 113, INDIA. Email: [email protected] August 25, 2003 Abstract We clarify the computational complexity of planarity testing, by showing that pla- narity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [Sak96]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FLSL. Previously, these problems were known to reside in the complexity class AC1,via a O(log n) time CRCW PRAM algorithm [RR94], although planarity checking for degree-three graphs had been shown to be in SL [Rei84, NTS95]. 1 Introduction The problem of determining if a graph is planar has been studied from several perspectives of algorithmic research. From most perspectives, optimal algorithms are already known. Linear-time sequential algorithms were presented by Hopcroft and Tarjan [HT74] and (via another approach) by combining the results of [LEC67, BL76, ET76]. In the context of ∗An extended abstract of this paper appeared in the Proceedings of STACS 2000 [AM00]. ySupported in part by NSF grants CCR-9734918 and and CCR-0104823. zPart of this work was done when this author was supported by the NSF grant CCR-9734918 on a visit to Rutgers University during summer 1999.
    [Show full text]
  • Planar and Grid Graph Reachability Problems
    Planar and Grid Graph Reachability Problems Eric Allender∗ David A. Mix Barrington† Tanmoy Chakraborty‡ Samir Datta§ Sambuddha Roy¶ Abstract We study the complexity of restricted versions of s-t-connectivity, which is the standard complete problem for NL. In particular, we focus on different classes of planar graphs, of which grid graphs are an important special case. Our main results are: • Reachability in graphs of genus one is logspace-equivalent to reachability in grid graphs (and in particular it is logspace-equivalent to both reachability and non-reachability in planar graphs). • Many of the natural restrictions on grid-graph reachability (GGR) are equivalent under AC0 reductions (for instance, undirected GGR, outdegree-one GGR, and indegree-one-outdegree-one GGR are all equivalent). These problems are all equivalent to the problem of determining whether a completed game position in HEX is a winning position, as well as to the problem of reachability in mazes studied by Blum and Kozen [BK78]. These problems provide natural examples of problems that are hard for NC1 under AC0 reductions but are not known to be hard for L; they thus give insight into the structure of L. • Reachability in layered planar graphs is logspace-equivalent to layered grid graph reachability (LGGR). We show that LGGR lies in UL (a subclass of NL). • Series-Parallel digraphs (on which reachability was shown to be decidable in logspace by Jakoby et al.) are a spe- cial case of single-source-single-sink planar directed acyclic graphs (DAGs); reachability for such graphs logspace reduces to single-source-single-sink acyclic grid graphs.
    [Show full text]
  • Planar Graph - Wikipedia, the Free Encyclopedia Page 1 of 7
    Planar graph - Wikipedia, the free encyclopedia Page 1 of 7 Planar graph From Wikipedia, the free encyclopedia In graph theory, a planar graph is a graph that can be Example graphs embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other Planar Nonplanar words, it can be drawn in such a way that no edges cross each other. A planar graph already drawn in the plane without edge intersections is called a plane graph or planar embedding of Butterfly graph the graph . A plane graph can be defined as a planar graph with a mapping from every node to a point in 2D space, and from K5 every edge to a plane curve, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Plane graphs can be encoded by combinatorial maps. It is easily seen that a graph that can be drawn on the plane can be drawn on the sphere as well, and vice versa. The equivalence class of topologically equivalent drawings on K3,3 the sphere is called a planar map . Although a plane graph has The complete graph external unbounded an or face, none of the faces of a planar K4 is planar map have a particular status. A generalization of planar graphs are graphs which can be drawn on a surface of a given genus. In this terminology, planar graphs have graph genus 0, since the plane (and the sphere) are surfaces of genus 0.
    [Show full text]
  • Planar Graphs, Planarity Testing and Embedding
    Planar Graphs, Planarity Testing and Embedding Planar Graphs, Planarity Testing and Embedding Swami Sarvottamananda Ramakrishna Mission Vivekananda University October 2014 1 / 151 Planar Graphs, Planarity Testing and Embedding OutlineI 1 Introduction Scope of the Lecture Definition of Planar Graphs Motivation for Planar Graphs Problem Definition 2 Characterisation of Planar Graphs Euler's Relation for the Planar Graphs Kuratowski's and Wagner's Theorems Proof of Kuratowski's theorem 3 Planarity Testing of Graphs Checking for Kuratowski's & Wagner's conditions Hopcroft and Tarjan's Algorithm Preliminaries Planarity Testing Algorithm 4 Planar Embedding of Graphs 2 / 151 Planar Graphs, Planarity Testing and Embedding OutlineII Embedding Phase of Planarity Algorithm Other Embeddings 5 Conclusion Open Problems Summary References 3 / 151 Planar Graphs, Planarity Testing and Embedding Introduction Introduction 4 / 151 Planarity Testing: Next, we give an algorithm to test planarity. Planar Embedding: Lastly we see how a given planar graph can be embedded in a plane. We also explore straight line embeddings. Planar Graphs, Planarity Testing and Embedding Introduction Scope Scope of the lecture Characterisation of Planar Graphs: First we introduce planar graphs, and give its characterisation and some simple properties. 5 / 151 Planar Embedding: Lastly we see how a given planar graph can be embedded in a plane. We also explore straight line embeddings. Planar Graphs, Planarity Testing and Embedding Introduction Scope Scope of the lecture Characterisation of Planar Graphs: First we introduce planar graphs, and give its characterisation and some simple properties. Planarity Testing: Next, we give an algorithm to test planarity. 6 / 151 Planar Graphs, Planarity Testing and Embedding Introduction Scope Scope of the lecture Characterisation of Planar Graphs: First we introduce planar graphs, and give its characterisation and some simple properties.
    [Show full text]