DOCSLIB.ORG

Planar Graph - Wikipedia, the Free Encyclopedia Page 1 of 7

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. See "graph embedding" for other related topics. Contents 1 Kuratowski's and Wagner's theorems 2 Other planarity criteria 2.1 Euler's formula 3 Related families of graphs 4 Other facts and definitions 5 Applications 6 Notes 7 References 8 External links Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem : A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the http://en.wikipedia.org/wiki/Planar_graph 5/23/2011 Planar graph - Wikipedia, the free encyclopedia Page 2 of 7 complete graph on five vertices ) or K3,3 ( complete bipartite graph on six vertices, three of which connect to each of the other three). A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •—— • to •—•—•) zero or more times. Equivalent formulations of this theorem, also known as "Theorem P" include A finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to K5 or K3,3 . In the Soviet Union, Kuratowski's theorem was known as the Pontryagin-Kuratowski theorem, as its proof was allegedly first given in Pontryagin's unpublished notes. By a long-standing academic tradition, such references are not taken into account in determining priority, so the Russian name of the theorem is not acknowledged internationally. Instead of considering subdivisions, Wagner's theorem deals with minors: A finite graph is planar if and only if it does not have K5 or K3,3 as a minor. An example of a graph which doesn't have K5 or K3,3 as its subgraph. However, it has a subgraph that is homeomorphic to K3,3 and is therefore not planar. Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now the Robertson-Seymour theorem, proved in a long series of papers. In the language of this theorem, K5 and K3,3 are the forbidden minors for the class of finite planar graphs. Other planarity criteria In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fast algorithms for this problem: for a graph with n vertices, it is possible to determine in time O(n) (linear time) whether the graph may be planar or not (see planarity testing ). http://en.wikipedia.org/wiki/Planar_graph 5/23/2011 Planar graph - Wikipedia, the free encyclopedia Page 3 of 7 For a simple, connected, planar graph with v vertices and e edges, the following simple planarity criteria hold: Theorem 1. If v ≥ 3 then e ≤ 3 v − 6; Theorem 2. If v > 3 and there are no cycles of length 3, then e ≤ 2 v − 4. In this sense, planar graphs are sparse graphs, in that they have only O( v) edges, asymptotically smaller than the maximum O An animation showing that the Petersen graph contains a minor isomorphic to the 2 K3,3 graph (v ). The graph K3,3 , for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. Note that these theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other methods may be used. For two planar graphs with v vertices, it is possible to determine in time O( v) whether they are isomorphic or not (see also graph isomorphism problem). [1] Whitney's planarity criterion gives a characterization based on the existence of an algebraic dual; MacLane's planarity criterion gives an algebraic characterization of finite planar graphs, via their cycle spaces; Fraysseix–Rosenstiehl's planarity criterion gives a characterization based on the existence of a bipartition of the cotree edges of a depth-first search tree. It is central to the left-right planarity testing algorithm ; Schnyder's theorem gives a characterization of planarity in terms of partial order dimension; Colin de Verdière's planarity criterion gives a characterization based on the maximum multiplicity of the second eigenvalue of certain Schrödinger operators defined by the graph. Euler's formula Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely-large region), then v − e + f = 2 . As an illustration, in the butterfly graph given above, v = 5, e = 6 and f = 3. If the second graph is http://en.wikipedia.org/wiki/Planar_graph 5/23/2011 Planar graph - Wikipedia, the free encyclopedia Page 4 of 7 redrawn without edge intersections, it has v = 4, e = 6 and f = 4. In general, if the property holds for all planar graphs of f faces, any change to the graph that creates an additional face while keeping the graph planar would keep v − e + f an invariant. Since the property holds for all graphs with f = 2, by mathematical induction it holds for all cases. Euler's formula can also be proven as follows: if the graph isn't a tree, then remove an edge which completes a cycle. This lowers both e and f by one, leaving v − e + f constant. Repeat until the remaining graph is a tree; trees have v = e + 1 and f = 1, yielding v − e + f = 2. i.e. the Euler characteristic is 2. In a finite, connected, simple , planar graph, any face (except possibly the outer one) is bounded by at least three edges and every edge touches at most two faces; using Euler's formula, one can then show that these graphs are sparse in the sense that e ≤ 3 v − 6 if v ≥ 3. A simple graph is called maximal planar if it is planar but adding any edge (on the given vertex set) would destroy that property. All faces (even the outer one) are then bounded by three edges, explaining the alternative terms triangular and triangulated for these graphs. If a triangular graph has v vertices with v > 2, then it has precisely 3 v − 6 edges and 2v − 4 faces. Euler's formula is also valid for simple polyhedra. This is no coincidence: every simple polyhedron can be turned into a The Goldner–Harary graph is maximal connected, simple, planar graph by using the polyhedron's planar. All its faces are bounded by three vertices as vertices of the graph and the polyhedron's edges as edges. edges of the graph. The faces of the resulting planar graph then correspond to the faces of the polyhedron. For example, the second planar graph shown above corresponds to a tetrahedron. Not every connected, simple, planar graph belongs to a simple polyhedron in this fashion: the trees do not, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra (equivalently: those formed from simple polyhedra) are precisely the finite 3-connected simple planar graphs. Related families of graphs Outerplanar graphs are graphs with an embedding in the plane such that all vertices belong to the unbounded face of the embedding. Every outerplanar graph is planar, but the converse is not true: K4 is planar but not outerplanar. A theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subdivision of K4 or of K2,3 . A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For k > 1 a planar embedding is k-outerplanar if removing the vertices on the outer face results in a ( k − 1)-outerplanar embedding. A graph is k-outerplanar if it has a k-outerplanar embedding An apex graph is a graph that may be made planar by the removal of one vertex, and a k-apex graph is a graph that may be made planar by the removal of at most k vertices.
Load more

Recommended publications
  • A Dynamic Data Structure for Planar Graph Embedding
    October 1987 UILU-ENG-87-2265 ACT-83 COORDINATED SCIENCE LABORATORY College of Engineering Applied Computation Theory A DYNAMIC DATA STRUCTURE FOR PLANAR GRAPH EMBEDDING Roberto Tamassia UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Approved for Public Release. Distribution Unlimited. UNCLASSIFIED___________ SEÒjrtlfy CLASSIFICATION OP THIS PAGE REPORT DOCUMENTATION PAGE 1 a. REPORT SECURITY CLASSIFICATION 1b. RESTRICTIVE MARKINGS Unclassified None 2a. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILABIUTY OF REPORT 2b. DECLASSIFICATION / DOWNGRADING SCHEDULE Approved for public release; distribution unlimited 4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S) UILU-ENG-87-2265 ACT #83 6a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION Coordinated Science Lab (If applicati!a) University of Illinois N/A National Science Foundation 6c ADDRESS (City, Statt, and ZIP Coda) 7b. ADDRESS (City, Stata, and ZIP Coda) 1101 W. Springfield Avenue 1800 G Street, N.W. Urbana, IL 61801 Washington, D.C. 20550 8a. NAME OF FUNDING/SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (If applicatila) National Science Foundation ECS-84-10902 8c. ADDRESS (City, Stata, and ZIP Coda) 10. SOURCE OF FUNDING NUMBERS 1800 G Street, N.W. PROGRAM PROJECT TASK WORK UNIT Washington, D.C. 20550 ELEMENT NO. NO. NO. ACCESSION NO. A Dynamic Data Structure for Planar Graph Embedding 12. PERSONAL AUTHOR(S) Tamassia, Roberto 13a. TYPE OF REPORT 13b. TIME COVERED 14-^ A T E OP REPORT (Year, Month, Day) 5. PAGE COUNT Technical FROM______ TO 1987, October 26 ? 41 16. SUPPLEMENTARY NOTATION 17. COSATI CODES 18. SUBJECT TERMS (Continua on ravarsa if nacassary and identify by block number) FIELD GROUP SUB-GROUP planar graph, planar embedding, dynamic data structure, on-line algorithm, analysis of algorithms present a dynamic data structure that allows for incrementally constructing a planar embedding of a planar graph.
    [Show full text]
  • Additive Non-Approximability of Chromatic Number in Proper Minor
    Additive non-approximability of chromatic number in proper minor-closed classes Zdenˇek Dvoˇr´ak∗ Ken-ichi Kawarabayashi† Abstract Robin Thomas asked whether for every proper minor-closed class , there exists a polynomial-time algorithm approximating the chro- G matic number of graphs from up to a constant additive error inde- G pendent on the class . We show this is not the case: unless P = NP, G for every integer k 1, there is no polynomial-time algorithm to color ≥ a K -minor-free graph G using at most χ(G)+ k 1 colors. More 4k+1 − generally, for every k 1 and 1 β 4/3, there is no polynomial- ≥ ≤ ≤ time algorithm to color a K4k+1-minor-free graph G using less than βχ(G)+(4 3β)k colors. As far as we know, this is the first non-trivial − non-approximability result regarding the chromatic number in proper minor-closed classes. We also give somewhat weaker non-approximability bound for K4k+1- minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with addi- tive error 6 under the additional assumption that the graph has no 4-cycles. arXiv:1707.03888v1 [cs.DM] 12 Jul 2017 The problem of determining the chromatic number, or even of just de- ciding whether a graph is colorable using a fixed number c 3 of colors, is NP-complete [7], and thus it cannot be solved in polynomial≥ time un- less P = NP.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • An Update on the Four-Color Theorem Robin Thomas
    thomas.qxp 6/11/98 4:10 PM Page 848 An Update on the Four-Color Theorem Robin Thomas very planar map of connected countries the five-color theorem (Theorem 2 below) and can be colored using four colors in such discovered what became known as Kempe chains, a way that countries with a common and Tait found an equivalent formulation of the boundary segment (not just a point) re- Four-Color Theorem in terms of edge 3-coloring, ceive different colors. It is amazing that stated here as Theorem 3. Esuch a simply stated result resisted proof for one The next major contribution came in 1913 from and a quarter centuries, and even today it is not G. D. Birkhoff, whose work allowed Franklin to yet fully understood. In this article I concentrate prove in 1922 that the four-color conjecture is on recent developments: equivalent formulations, true for maps with at most twenty-five regions. The a new proof, and progress on some generalizations. same method was used by other mathematicians to make progress on the four-color problem. Im- Brief History portant here is the work by Heesch, who developed The Four-Color Problem dates back to 1852 when the two main ingredients needed for the ultimate Francis Guthrie, while trying to color the map of proof—“reducibility” and “discharging”. While the the counties of England, noticed that four colors concept of reducibility was studied by other re- sufficed. He asked his brother Frederick if it was searchers as well, the idea of discharging, crucial true that any map can be colored using four col- for the unavoidability part of the proof, is due to ors in such a way that adjacent regions (i.e., those Heesch, and he also conjectured that a suitable de- sharing a common boundary segment, not just a velopment of this method would solve the Four- point) receive different colors.
    [Show full text]
  • Bandwidth, Expansion, Treewidth, Separators and Universality for Bounded-Degree Graphs$
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 31 (2010) 1217–1227 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphsI Julia Böttcher a, Klaas P. Pruessmann b, Anusch Taraz a, Andreas Würfl a a Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85747 Garching bei München, Germany b Institute for Biomedical Engineering, University and ETH Zurich, Gloriastr. 35, 8092, Zürich, Switzerland article info a b s t r a c t Article history: We establish relations between the bandwidth and the treewidth Received 3 March 2009 of bounded degree graphs G, and relate these parameters to the size Accepted 13 October 2009 of a separator of G as well as the size of an expanding subgraph of Available online 24 October 2009 G. Our results imply that if one of these parameters is sublinear in the number of vertices of G then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each γ > 0 every n-vertex graph with minimum degree 3 C . 4 γ /n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large. ' 2009 Elsevier Ltd.
    [Show full text]
  • [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]
  • Some Conjectures in Chromatic Topological Graph Theory Joan P
    Some Conjectures in Chromatic Topological Graph Theory Joan P. Hutchinson Macalester College, emerita Thanks to Stan Wagon for the colorful graphics. Ñ 1 2 3 4 5 6 7 ALMOST FOUR-COLORING ON SURFACES: ALMOST FOUR-COLORING ON SURFACES: #1. M.O. Albertson’s favorite CONJECTURE (1980): All vertices of a triangulation of the torus can be 4-col- ored except for at most three vertices. Motivation: Every graph on the torus can be 7-colored, and every 7-chromatic toroidal graph contains K7, a trian- gulation of the torus. Every 6-chromatic toroidal graph contains one of four graphs. [Thomassen 1994] The list for 5-chromatic toroidal graphs is infinite... An affirmative answer to this conjecture implies the Four Color Theorem. In Jensen & Toft, Graph Coloring Problems, Albertson’s Four-Color Problem. CONJECTURE: For every surface S, there is an integer f HSL such that all but f HSL vertices of a graph embed- dable on S can be 4-colored. TWO & THREE-COLORING ON SURFACES #2. The “easiest” coloring result states that every plane graph can be two-colored provided every face is bounded by an even number of edges. What is true on surfaces? Locally bipartite (aka evenly embedded) graphs are those embedded on a surface with all faces bounded by an even number of edges. There is a Heawood/Ringel type theorem for locally bipar- tite graphs on surfaces. The more “modern” question asks about locally planar, locally bipartite graphs on surfaces: The more “modern” question asks about locally planar, locally bipartite graphs on surfaces: Locally planar (Albertson & Stromquist 1980) means that all noncontractible cycles are long, as long as needed for the conjecture/question/theorem at hand.
    [Show full text]
  • On a Conjecture Concerning the Petersen Graph
    On a conjecture concerning the Petersen graph Donald Nelson Michael D. Plummer Department of Mathematical Sciences Department of Mathematics Middle Tennessee State University Vanderbilt University Murfreesboro,TN37132,USA Nashville,TN37240,USA [email protected] [email protected] Neil Robertson* Xiaoya Zha† Department of Mathematics Department of Mathematical Sciences Ohio State University Middle Tennessee State University Columbus,OH43210,USA Murfreesboro,TN37132,USA [email protected] [email protected] Submitted: Oct 4, 2010; Accepted: Jan 10, 2011; Published: Jan 19, 2011 Mathematics Subject Classifications: 05C38, 05C40, 05C75 Abstract Robertson has conjectured that the only 3-connected, internally 4-con- nected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover, this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumption of internal 4-connectivity cannot be dropped as an hypothesis in the original conjecture. We then summarize our results aimed toward the solution of the conjec- ture in its original form. In particular, let G be any 3-connected internally-4- connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord. If C is any girth cycle in G then N(C)\V (C) cannot be edgeless, and if N(C)\V (C) contains a path of length at least 2, then the conjecture is true. Consequently, if the conjecture is false and H is a counterexample, then for any girth cycle C in H, N(C)\V (C) induces a nontrivial matching M together with an independent set of vertices.
    [Show full text]
  • Nowhere-Zero Flows in Regular Matroids and Hadwiger's Conjecture
    NOWHERE-ZERO FLOWS IN REGULAR MATROIDS AND HADWIGER'S CONJECTURE LUIS A. GODDYN AND WINFRIED HOCHSTATTLER¨ To the memory of Reinhard B¨orger Abstract. We present a tool that shows, that the existence of a k-nowhere-zero-flow is compatible with 1-,2- and 3-sums in regular matroids. As application we present a conjecture for regular matroids that is equivalent to Hadwiger's conjecture for graphs and Tuttes's 4- and 5-flow conjectures. Keywords: nowhere zero flow, regular matroid, chromatic number, flow number, total unimodularity 1. Introduction A (real) matrix is totally unimodular (TUM) if each subdeterminant belongs to f0; ±1g. Totally unimodular matrices enjoy several nice properties which give them a fundamental role in combinatorial optimization and matroid theory. In this note we prove that the TUM possesses an attractive property. Let S ⊆ R, and let A be a real matrix. A column vector f is a S-flow of A if Af = 0 and every entry of f is a member of ±S. For any additive abelian group Γ use the notation Γ∗ = Γ n f0g. For a TUM A and a column vector f with entries in Γ, the product Af is a well defined column vector with entries in Γ, by interpreting (−1)γ to be the additive inverse of γ. It is convenient to use the language of matroids. A regular oriented matroid M is an oriented matroid that is representable M = M[A] by a TUM matrix A. Here the elements E(M) of M label the columns of A. Each (signed) cocircuit D = (D+;D−) of M corresponds to a f0; ±1g-valued vector in the row space of A and having minimal support.
    [Show full text]
  • Representing Graphs by Polygons with Edge Contacts in 3D
    Representing Graphs by Polygons with Side Contacts in 3D∗ Elena Arseneva1, Linda Kleist2, Boris Klemz3, Maarten Löffler4, André Schulz5, Birgit Vogtenhuber6, and Alexander Wolff7 1 Saint Petersburg State University, Russia [email protected] 2 Technische Universität Braunschweig, Germany [email protected] 3 Freie Universität Berlin, Germany [email protected] 4 Utrecht University, the Netherlands [email protected] 5 FernUniversität in Hagen, Germany [email protected] 6 Technische Universität Graz, Austria [email protected] 7 Universität Würzburg, Germany orcid.org/0000-0001-5872-718X Abstract A graph has a side-contact representation with polygons if its vertices can be represented by interior-disjoint polygons such that two polygons share a common side if and only if the cor- responding vertices are adjacent. In this work we study representations in 3D. We show that every graph has a side-contact representation with polygons in 3D, while this is not the case if we additionally require that the polygons are convex: we show that every supergraph of K5 and every nonplanar 3-tree does not admit a representation with convex polygons. On the other hand, K4,4 admits such a representation, and so does every graph obtained from a complete graph by subdividing each edge once. Finally, we construct an unbounded family of graphs with average vertex degree 12 − o(1) that admit side-contact representations with convex polygons in 3D. Hence, such graphs can be considerably denser than planar graphs. 1 Introduction A graph has a contact representation if its vertices can be represented by interior-disjoint geometric objects1 such that two objects touch exactly if the corresponding vertices are adjacent.
    [Show full text]
  • INTRINSIC LINKING in DIRECTED GRAPHS 1. Introduction Research
    INTRINSIC LINKING IN DIRECTED GRAPHS JOEL STEPHEN FOISY, HUGH NELSON HOWARDS, NATALIE ROSE RICH* Abstract. We extend the notion of intrinsic linking to directed graphs. We give methods of constructing intrinsically linked directed graphs, as well as complicated directed graphs that are not intrinsically linked. We prove that the double directed version of a graph G is intrinsically linked if and only if G is intrinsically linked. One Corollary is that J6, the complete symmetric directed graph on 6 vertices (with 30 directed edges), is intrinsically linked. We further extend this to show that it is possible to find a subgraph of J6 by deleting 6 edges that is still intrinsically linked, but that no subgraph of J6 obtained by deleting 7 edges is intrinsically linked. We also show that J6 with an arbitrary edge deleted is intrinsically linked, but if the wrong two edges are chosen, J6 with two edges deleted can be embedded linklessly. 1. Introduction Research in spatial graphs has been rapidly on the rise over the last fif- teen years. It is interesting because of its elegance, depth, and accessibility. Since Conway and Gordon published their seminal paper in 1983 [4] show- ing that the complete graph on 6 vertices is intrinsically linked (also shown independently by Sachs [10]) and the complete graph on 7 vertices is in- trinsically knotted, over 62 papers have been published referencing it. The topology of graphs is, of course, interesting because it touches on chem- istry, networks, computers, etc. This paper takes the natural step of asking about the topology of directed graphs, sometimes called digraphs.
    [Show full text]