Restricted String Representations
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On Grounded L-Graphs and Their Relatives Arxiv:1808.04148V1
On grounded L-graphs and their relatives Vít Jelínek∗ Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czechia [email protected] Martin Töpfery Institute of Science and Technology, Klosterneuburg, Austria [email protected] August 14, 2018 Abstract We consider the graph classes Grounded-L and Grounded-fL; L g corresponding to graphs that admit an intersection representation by L-shaped curves (or L-shaped and L -shaped curves, respectively), where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare these classes to related intersection classes, such as the grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give con- structions showing that these classes are all distinct and satisfy only arXiv:1808.04148v1 [math.CO] 13 Aug 2018 trivial or previously known inclusions. ∗Supported by the project 16-01602Y of the Czech Science Foundation and by project Neuron Impuls of the Neuron Fund for Support of Science. yThis project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385. 1 1 Introduction An intersection representation of a graph G = (V; E) is a map that assigns to every vertex x 2 V a set sx in such a way that two vertices x and y are adjacent if and only if the two corresponding sets sx and sy intersect. -
Separators in Region Intersection Graphs
Separators in region intersection graphs James R. Lee∗ Abstract For undirected graphs G V; E and G0 V0; E0 , say that G is a region intersection ¹ º ¹ º graph over G0 if there is a family of connected subsets Ru V0 : u V of G0 such that f ⊆ 2 g u; v E Ru Rv , . f g 2 () \ ; We show if G0 excludes the complete graph Kh as a minor for some h > 1, then every region intersection graph G over G0 with m edges has a balanced separator with at most chpm nodes, where ch is a constant depending only on h. If G additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. String graphs are precisely region intersection graphs over planar graphs. Thus the preceding result implies that every string graph with m edges has a balanced separator of size O pm . This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture¹ º of Fox and Pach (2010), and improves over the O pm log m bound of Matoušek (2013). ¹ º Contents 1 Introduction 2 1.1 Balanced separators and extremal spread.........................3 1.2 Eigenvalues and L2-extremal spread............................7 1.3 Additional applications....................................7 1.4 Preliminaries..........................................7 2 Vertex separators and conformal graph metrics8 2.1 Conformal graphs.......................................8 2.2 Padded partitions and random separators......................... 10 3 Multi-flows, congestion, and crossings 12 3.1 Duality between conformal metrics and multi-flows.................. -
Spectral Conditions for Graph Rigidity in the Euclidean Plane
Spectral conditions for graph rigidity in the Euclidean plane Sebastian M. Cioab˘a∗ and Xiaofeng Guy January 20, 2020 Abstract Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair (G; p) of graph G together with a map p of the vertices of G into the Euclidean d-space. We view the edges of (G; p) as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combina- torial characterization of rigid graphs in the Euclidean plane. In 1982, Lov´aszand Yemini discovered a new characterization and proved that every 6-connected graph is rigid in the Euclidean plane. Consequently, if Fiedler's algebraic connectivity is at least 6, then G is rigid. In this paper, we show that if G has minimum degree δ ≥ 6 and algebraic connectivity 1 greater than 2 + δ−1 , then G is rigid. We prove a more general result giving a necessary spectral condition for packing k edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains k edge-disjoint 2-connected spanning subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees. MSC: 05C50, 05C70 Key words: eigenvalue, algebraic connectivity, rigid graph, connectivity 1 Introduction In this paper, we consider finite undirected simple graphs. -
Geometric Intersection Patterns and the Theory of Topological Graphs
Geometric Intersection Patterns and the Theory of Topological Graphs J´anosPach∗ Abstract. The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3-dimensional convex polytopes. The analogous statement is false in any fixed dimension if the polytopes are allowed to have only a bounded number of faces or are replaced by simple geometric objects that can be described in terms of a bounded number of real parameters. Intersection graphs of various classes of geometric objects, even in the plane, have interesting structural and extremal properties. We survey problems and results on geometric intersection graphs and, more gener- ally, intersection patterns. Many of the questions discussed were originally raised by Berge, Erd}os,Gr¨unbaum, Hadwiger, Tur´an,and others in the context of classical topol- ogy, graph theory, and combinatorics (related, e.g., to Helly's theorem, Ramsey theory, perfect graphs). The rapid development of computational geometry and graph drawing algorithms in the last couple of decades gave further impetus to research in this field. A topological graph is a graph drawn in the plane so that its vertices are represented by points and its edges by possibly intersecting simple continuous curves connecting the corresponding point pairs. We give applications of the results concerning intersection patterns in the theory of topological graphs. Mathematics Subject Classification (2010). Primary 05C35; Secondary 05C62, 52C10. -
From Planar Graphs to Higher Dimension Lucas Isenmann
From planar graphs to higher dimension Lucas Isenmann To cite this version: Lucas Isenmann. From planar graphs to higher dimension. Discrete Mathematics [cs.DM]. Université Montpellier, 2019. English. NNT : 2019MONTS142. tel-02931761 HAL Id: tel-02931761 https://tel.archives-ouvertes.fr/tel-02931761 Submitted on 7 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE POUR OBTENIR LE GRADE DE DOCTEUR DE L’UNIVERSITÉ DE MONTPELLIER En informatique École doctorale I2S Unité de recherche LIRMM UMR 5506 Des graphes planaires vers des dimensions supérieures Présentée par Lucas Isenmann Décembre 2019 Sous la direction de Daniel GONÇALVES Devant le jury composé de Jean CARDINAL, Professeur, Université Libre de Bruxelles Rapporteur Nicolas BONICHON, Maitre de conférences, Université de Bordeaux Examinateur William PUECH, Professeur des universités, Université de de Montpellier Examinateur Nicolas TROTIGNON, Directeur de recherche, ENS de Lyon Examinateur Contents 1 Dushnik-Miller dimension of some geometric complexes 17 1.1 Introduction . 17 1.1.1 Basic notions . 17 1.1.2 Delaunay graphs and their variants . 19 1.1.3 Dushnik-Miller dimension and representations . 22 1.1.4 State of the art . 26 1.1.5 Organization of the chapter . -
On the Chromatic Number of Disjointness Graphs of Curves
On the Chromatic Number of Disjointness Graphs of Curves János Pach École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Rényi Institute, Budapest, Hungary janos.pach@epfl.ch István Tomon École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland istvan.tomon@epfl.ch Abstract Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis. We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that k+1 ω(G) = k, then χ(G) ≤ 2 . If we only require that every curve is x-monotone and intersects the k+1 k+2 y-axis, then we have χ(G) ≤ 2 3 . Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist Kk-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k4) colors. This matches the upper bound up to a constant factor. 2012 ACM Subject Classification Mathematics of computing → Graph theory Keywords and phrases string graph, chromatic number, intersection graph Digital Object Identifier 10.4230/LIPIcs.SoCG.2019.54 Related Version https://arxiv.org/abs/1811.09158 Funding János Pach: Research partially supported by Swiss National Science Foundation grants no. -
Some Inequalities for the Tutte Polynomial
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 32 (2011) 422–433 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Some inequalities for the Tutte polynomial Laura E. Chávez-Lomelí a, Criel Merino b,1, Steven D. Noble c, Marcelino Ramírez-Ibáñez b a Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Avenida San Pablo 180, colonia Reynosa Tamaulipas, Delegación Azcapotzalco, Mexico b Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de la Investigación Científica, Circuito Exterior, C.U. Coyoacán 04510, México D.F., Mexico c Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge UB8 3PH, UK article info a b s t r a c t Article history: We prove that the Tutte polynomial of a coloopless paving matroid Received 22 February 2010 is convex along the portion of the line x C y D p lying in the Accepted 6 September 2010 positive quadrant. Every coloopless paving matroid is in the class Available online 29 December 2010 of matroids which contain two disjoint bases or whose ground set is the union of two bases. For this latter class we give a proof that TM .a; a/ ≤ maxfTM .2a; 0/; TM .0; 2a/g for a ≥ 2. We conjecture that TM .1; 1/ ≤ maxfTM .2; 0/; TM .0; 2/g for the same class of matroids. We also prove this conjecture for some families of graphs and matroids. ' 2010 Elsevier Ltd. All rights reserved. 1. Introduction The Tutte polynomial is a two-variable polynomial which can be defined for a graph G or more generally a matroid M. -
Separators in Region Intersection Graphs
Separators in Region Intersection Graphs James R. Lee Computer Science, University of Washington, Seattle, USA [email protected] Abstract For undirected graphs G = (V, E) and G0 = (V0,E0), say that G is a region intersection graph over G0 if there is a family of connected subsets {Ru ⊆ V0 : u ∈ V } of G0 such that {u, v} ∈ E ⇐⇒ Ru ∩ Rv 6= ∅. We show if G excludes the complete graph K as a minor for some h ≥ 1, then every region 0 h √ intersection graph G over G0 with m edges has a balanced separator with at most ch m nodes, where ch is a constant depending only on h. If G additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. String graphs are precisely region intersection graphs over planar graphs. Thus the preceding result implies that √ every string graph with m edges has a balanced separator of size O( m). This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010), √ and improves over the O( m log m) bound of Matoušek (2013). 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.1.6 Optim- ization, G.2.2 Graph Theory Keywords and phrases Graph separators, planar graphs, spectral partitioning Digital Object Identifier 10.4230/LIPIcs.ITCS.2017.1 1 Introduction Consider an undirected graph G0 = (V0,E0). A graph G = (V, E) is said to be a region intersection graph (rig) over G0 if the vertices of G correspond to connected subsets of G0 and there is an edge between two vertices of G precisely when those subsets intersect. -
BIPARTITE RIGIDITY 1. Introduction 1.1. Three Basic Properties Of
BIPARTITE RIGIDITY GIL KALAI, ERAN NEVO, AND ISABELLA NOVIK Abstract. We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k; l the notions of (k; l)-rigid and (k; l)-stress free bipartite graphs. This theory coincides with the study of Babson{Novik's balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, and gluing lemmas, and apply these results to derive a bipartite analog of the rigidity criterion for planar graphs. Our result asserts that for a planar bipartite graph G its balanced shifting, Gb, does not contain K3;3; equivalently, planar bipartite graphs are generically (2; 2)-stress free. We also discuss potential applications of this theory to Jockusch's cubical lower bound conjecture and to upper bound conjectures for embedded simplicial complexes. 1. Introduction 1.1. Three basic properties of planar graphs. We start with three important results on planar graphs and a motivating conjecture in a higher dimension: Proposition 1.1 (Euler, Descartes). A simple planar graph with n ≥ 3 vertices has at most 3n − 6 edges. Proposition 1.2 (Wagner, Kuratowski (easy part)). A planar graph does not contain K5 and K3;3 as minors. The first result (that can be traced back to Descartes) is a simple consequence of Euler's theorem. The second result is the easy part of Wagner's characterization of planar graphs [41] which asserts that not having K5 and K3;3 as minors characterizes planarity. We will now state a third fundamental result on planar graphs. -
A Separator Theorem for String Graphs and Its Applications
A separator theorem for string graphs and its applications Jacob Fox¤ J¶anosPachy Abstract A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the p removal of O(m3=4 log m) vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt;t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any ² > 0, there is an integer g(²) such that every string graph with n vertices and girth at least g(²) has at most (1 + ²)n edges. Furthermore, the number of such graphs is at most (1 + ²)nT (n), where T (n) = nn¡2 is the number of labeled trees on n vertices. 1 Introduction A large part of computational geometry deals with representation and manipulation of various geomet- ric objects. Special attention is paid to pairs of objects that are in contact with each other: detecting intersections among line segments, for example, belongs to the oldest and best studied chapter of com- putational geometry, already addressed in the ¯rst monograph devoted to the subject [37]. Yet, even in the special case of segments, little is known about elementary structural properties of the arising intersection patterns. The recognition of such intersection patterns (intersection graphs) is known to be NP-hard [21], [22]. -
Planar Minimally Rigid Graphs and Pseudo-Triangulations
Planar Minimally Rigid Graphs and Pseudo-Triangulations Ruth Haas 1 David Orden 2 G¨unter Rote 3 Francisco Santos 2 Brigitte Servatius 4 Herman Servatius 4 Diane Souvaine 5 Ileana Streinu 6 and Walter Whiteley 7 Abstract Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than π). In this paper we prove that the opposite state- ment is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embed- ding algorithms. They also provide - to the best of our knowledge - the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces. These constraints are described by combinatorial pseudo-triangulations, first defined and studied in this paper. Also of interest are our two proof techniques, one based on Henneberg inductive constructions from combinatorial rigidity theory, the other on a generalization of Tutte’s barycentric embeddings to directed graphs. 1 Introduction In this paper we bring together two classical topics in graph theory, planarity and rigidity, to answer the fundamental question (posed in [42]) of characterizing the class of planar graphs which admit pointed pseudo-triangular embeddings. Our main result is that this coincides with the class of all planar minimally rigid graphs (planar Laman graphs). Furthermore we extend the result in several directions, attacking the same type of question for other (not necessarily pointed) classes of pseudo-triangulations and for combinatorial pseudo-triangulations, a new class of objects first introduced and studied in this paper. -
Notes on Graph Product Structure Theory
Notes on Graph Product Structure Theory Zdenekˇ Dvorák,ˇ Tony Huynh, Gwenaël Joret, Chun-Hung Liu, and David R. Wood Abstract It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys gener- alisations of this result for graphs on surfaces, minor-closed classes, various non- minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph prod- uct structure theorems for various geometrically defined graph classes, and present several open problems. Zdenekˇ Dvorákˇ Charles University, Prague, Czech Republic Supported by project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foun- dation e-mail: [email protected] Tony Huynh and David R. Wood School of Mathematics, Monash University, Melbourne, Australia Research supported by the Australian Research Council e-mail: [email protected],[email protected] Gwenaël Joret Département d’Informatique, Université libre de Bruxelles, Brussels, Belgium Research supported by the Wallonia-Brussels Federation of Belgium, and by the Australian Re- search Council e-mail: [email protected] Chun-Hung Liu Department of Mathematics, Texas A&M University, College Station, Texas, USA Partially supported by the NSF under Grant No. DMS-1929851 e-mail: [email protected] 1 2 Zdenekˇ Dvorák,ˇ Tony Huynh, Gwenaël Joret, Chun-Hung Liu, and David R.