An Annotated Bibliography on 1-Planarity
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[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]). -
Constrained Representations of Map Graphs and Half-Squares
Constrained Representations of Map Graphs and Half-Squares Hoang-Oanh Le Berlin, Germany [email protected] Van Bang Le Universität Rostock, Institut für Informatik, Rostock, Germany [email protected] Abstract The square of a graph H, denoted H2, is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B = (X, Y, EB ) are the subgraphs of B2 induced by the color classes X and Y , B2[X] and B2[Y ]. For a given graph 2 G = (V, EG), if G = B [V ] for some bipartite graph B = (V, W, EB ), then B is a representation of G and W is the set of points in B. If in addition B is planar, then G is also called a map graph and B is a witness of G [Chen, Grigni, Papadimitriou. Map graphs. J. ACM, 49 (2) (2002) 127-138]. While Chen, Grigni, Papadimitriou proved that any map graph G = (V, EG) has a witness with at most 3|V | − 6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown. We also consider half-squares of tree-convex bipartite graphs and prove the following complexity 2 dichotomy: Given a graph G = (V, EG) and an integer k, deciding if G = B [V ] for some tree-convex bipartite graph B = (V, W, EB ) with |W | ≤ k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise. -
New Bounds on the Number of Edges in a K-Map Graph
New bounds on the edge number of a k-map graph ∗ Zhi-Zhong Chen y Abstract It is known that for every integer k 4, each k-map graph with n vertices has at most ≥ kn 2k edges. Previously, it was open whether this bound is tight or not. We show that this − bound is tight for k = 4, 5. We also show that this bound is not tight for large enough k (namely, k 374); more precisely, we show that for every 0 < < 3 and for every integer k 140 , ≥ 328 ≥ 41 each k-map graph with n vertices has at most ( 325 + )kn 2k edges. This result implies the 328 − first polynomial (indeed linear) time algorithm for coloring a given k-map graph with less than 2k colors for large enough k. We further show that for every positive multiple k of 6, there are 11 1 infinitely many integers n such that some k-map graph with n vertices has at least ( 12 k + 3 )n edges. Key words. Planar graph, plane embedding, sphere embedding, map graph, k-map graph, edge number. AMS subject classifications. 05C99, 68R05, 68R10. Abbreviated title. Edge Number of k-Map Graphs. 1 Introduction Chen, Grigni, and Papadimitriou [6] studied a modified notion of planar duality, in which two nations of a (political) map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planar duality requires). Such adjacencies define a map graph. The map graph is called a k-map graph for some positive integer k, if no more than k nations on the map meet at a point. -
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. -
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. -
The Bidimensionality Theory and Its Algorithmic
The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi B.S., Sharif University of Technology, 2000 M.S., University of Waterloo, 2001 Submitted to the Department of Mathematics in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 °c MohammadTaghi Hajiaghayi, 2005. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Mathematics April 29, 2005 Certi¯ed by. Erik D. Demaine Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Rodolfo Ruben Rosales Chairman, Applied Mathematics Committee Accepted by . Pavel I. Etingof Chairman, Department Committee on Graduate Students 2 The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi Submitted to the Department of Mathematics on April 29, 2005, in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY Abstract Our newly developing theory of bidimensional graph problems provides general techniques for designing e±cient ¯xed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k £ k grid graph (and similar graphs) grows with k, typically as (k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). -
Layouts of Expander Graphs
CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2016, Article 1, pages 1–21 http://cjtcs.cs.uchicago.edu/ Layouts of Expander Graphs † ‡ Vida Dujmovic´∗ Anastasios Sidiropoulos David R. Wood Received January 20, 2015; Revised November 26, 2015, and on January 3, 2016; Published January 14, 2016 Abstract: Bourgain and Yehudayoff recently constructed O(1)-monotone bipartite ex- panders. By combining this result with a generalisation of the unraveling method of Kannan, we construct 3-monotone bipartite expanders, which is best possible. We then show that the same graphs admit 3-page book embeddings, 2-queue layouts, 4-track layouts, and have simple thickness 2. All these results are best possible. Key words and phrases: expander graph, monotone layout, book embedding, stack layout, queue layout, track layout, thickness 1 Introduction Expanders are classes of highly connected graphs that are of fundamental importance in graph theory, with numerous applications, especially in theoretical computer science [31]. While the literature contains various definitions of expanders, this paper focuses on bipartite expanders. For e (0;1], a bipartite 2 graph G with bipartition V(G) = A B is a bipartite e-expander if A = B and N(S) > (1 + e) S for A [ j j j j j j j j every subset S A with S j j . Here N(S) is the set of vertices adjacent to some vertex in S. An infinite ⊂ j j 6 2 family of bipartite e-expanders, for some fixed e > 0, is called an infinite family of bipartite expanders. There has been much research on constructing and proving the existence of expanders with various desirable properties. -
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. -
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs∗
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs∗ Sariel Har-Peledy Kent Quanrudz September 23, 2015 Abstract We investigate the family of intersection graphs of low density objects in low dimensional Eu- clidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion. We present efficient (1 + ")-approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTAS's for these problems are known for subclasses of this graph family. These results have immediate interesting applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow). We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth. 1. Introduction Many classical optimization problems are intractable to approximate, let alone solve. Motivated by the discrepancy between the worst-case analysis and real-world success of algorithms, more realistic models of input have been developed, alongside algorithms that take advantage of their properties. In this paper, we investigate approximability of some classical optimization problems (e.g., set cover and independent set, among others) for two closely-related families of graphs: Graphs with polynomially- bounded expansion, and intersection graphs of geometric objects with low-density. -
Characterising Bounded Expansion by Neighbourhood Complexity
Characterising Bounded Expansion by Neighbourhood Complexity Felix Reidl Fernando S´anchez Villaamil Konstantinos Stavropoulos RWTH Aachen University freidl,fernando.sanchez,[email protected]. Abstract We show that a graph class G has bounded expansion if and only if it has bounded r-neighbourhood complexity, i.e. for any vertex set X of any subgraph H of G ∈ G, the number of subsets of X which are exact r-neighbourhoods of vertices of H on X is linear in the size of X. This is established by bounding the r-neighbourhood complexity of a graph in terms of both its r-centred colouring number and its weak r-colouring number, which provide known characterisations to the property of bounded expansion. 1 Introduction Graph classes of bounded expansion (and their further generalisation, nowhere dense classes) have been introduced by Neˇsetˇriland Ossona de Mendez [23, 24, 25] as a general model of structurally sparse graph classes. They include and generalise many other natural sparse graph classes, among them all classes of bounded degree, classes of bounded genus, and classes defined by arXiv:1603.09532v2 [cs.DM] 2 Nov 2016 excluded (topological) minors. Nowhere dense classes even include classes that locally exclude a minor, which in turn generalises graphs with locally bounded treewidth. The appeal of this notion and its applications stems from the fact that bounded expansion has turned out to be a very robust property of graph classes with various seemingly unrelated characterisations (see [19, 25]). These include characterisations through the density of shallow minors [23], quasi-wideness [3] low treedepth colourings [23], and generalised colouring numbers [31]. -
Applications of Bipartite Matching to Problems in Object Recognition
Applications of Bipartite Matching to Problems in Object Recognition Ali Shokoufandeh Sven Dickinson Department of Mathematics and Department of Computer Science and Computer Science Center for Cognitive Science Drexel University Rutgers University Philadelphia, PA New Brunswick, NJ Abstract The matching of hierarchical (e.g., multiscale or multilevel) image features is a common problem in object recognition. Such structures are often represented as trees or directed acyclic graphs, where nodes represent image feature abstractions and arcs represent spatial relations, mappings across resolution levels, component parts, etc. Such matching problems can be formulated as largest isomorphic subgraph or largest isomorphic subtree problems, for which a wealth of literature exists in the graph algorithms community. However, the nature of the vision instantiation of this problem often precludes the direct application of these methods. Due to occlusion and noise, no significant isomorphisms may exists between two graphs or trees. In this paper, we review our application of a more general class of matching methods, called bipartite matching, to two problems in object recognition. 1 Introduction The matching of hierarchical (e.g., multiscale or multilevel) image features is a common problem in object recognition. Such structures are often represented as trees or directed acyclic graphs, where nodes represent image feature abstractions and arcs represent spatial relations, mappings across resolution levels, compo- nent parts, etc. The requirements of matching include computing a correspondence between nodes in an image structure and nodes in a model structure, as well as computing an overall measure of distance (or, alternatively, similarity) between the two structures. Such matching problems can be formulated as largest isomorphic subgraph or largest isomorphic subtree problems, for which a wealth of literature exists in the graph algorithms community. -
Lecture Beyond Planar Graphs 1 Overview 2 Preliminaries
Approximation Algorithms Workshop June 17, 2011, Princeton Lecture Beyond Planar Graphs Erik Demaine Scribe: Siamak Tazari 1 Overview In the last lecture we learned about approximation schemes in planar graphs. In this lecture we go beyond planar graphs and consider bounded-genus graphs, and more generally, minor-closed graph classes. In particular, we discuss the framework of bidimensionality that can be used to obtain approximation schemes for many problems in minor-closed graph classes. Afterwards, we consider decomposition techniques that can be applied to other types of problems. The main motivation behind this line of work is that some networks are not planar but conform to some generalization of the concept of planarity, like being embeddable on a surface of bounded genus. Furthermore, by Kuratowski's famous theorem [Kur30], planarity can be characterized be excluding K5 and K3;3 as minors. Hence, it seems natural to study further classes with excluded minors as a generalization of planar graphs. One of the goals of this research is to identify the largest classes of graphs for which we can still ob- tain good and efficient approximations, i.e. to find out how far beyond planarity we can go. Natural candidate classes are graphs of bounded genus, graphs excluding a fixed minor, powers thereof, but also further generalizations, like odd-minor-free graphs, graphs of bounded expansion and nowhere dense classes of graphs. Another objective is to derive general approximation frameworks, two of which we are going to discuss here: bidimensionality and contraction decomposition. The main tools we are going to use come on one hand from graph structure theory and on the other hand from algorithmic results.