4. Elementary Graph Algorithms in External Memory∗
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Arxiv:1908.08938V1 [Cs.DS] 23 Aug 2019 U ≺ W ≺ X ≺ V (Under the Same Assumptions As Above)
Mixed Linear Layouts: Complexity, Heuristics, and Experiments Philipp de Col, Fabian Klute, and Martin N¨ollenburg Algorithms and Complexity Group, TU Wien, Vienna, Austria [email protected],ffklute,[email protected] Abstract. A k-page linear graph layout of a graph G = (V; E) draws all vertices along a line ` and each edge in one of k disjoint halfplanes called pages, which are bounded by `. We consider two types of pages. In a stack page no two edges should cross and in a queue page no edge should be nested by another edge. A crossing (nesting) in a stack (queue) page is called a conflict. The algorithmic problem is twofold and requires to compute (i) a vertex ordering and (ii) a page assignment of the edges such that the resulting layout is either conflict-free or conflict-minimal. While linear layouts with only stack or only queue pages are well-studied, mixed s-stack q-queue layouts for s; q ≥ 1 have received less attention. We show NP-completeness results on the recognition problem of certain mixed linear layouts and present a new heuristic for minimizing conflicts. In a computational experiment for the case s; q = 1 we show that the new heuristic is an improvement over previous heuristics for linear layouts. 1 Introduction Linear graph layouts, in particular book embeddings [1,11] (also known as stack layouts) and queue layouts [8,9], form a classic research topic in graph drawing with many applications beyond graph visualization as surveyed by Dujmovi´cand Wood [5]. A k-page linear layout Γ = (≺; P) of a graph G = (V; E) consists of an order ≺ on the vertex set V and a partition of E into k subsets P = fP1;:::;Pkg called pages. -
Planar Graphs Have Bounded Queue-Number
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) Planar Graphs have Bounded Queue-Number Vida Dujmovic´∗, Gwenaël Joret†, Piotr Micek‡, Pat Morin§, Torsten Ueckerdt¶, David R. Wood ∗ School of Computer Science and Electrical Eng., University of Ottawa, Canada ([email protected]) † Département d’Informatique, Université Libre de Bruxelles, Brussels, Belgium ([email protected]) ‡ Faculty of Mathematics and Computer Science, Jagiellonian University, Poland ([email protected]) § School of Computer Science, Carleton University, Ottawa, Canada ([email protected]) ¶ Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany ([email protected]) School of Mathematics, Monash University, Melbourne, Australia ([email protected]) Abstract—We show that planar graphs have bounded queue- open whether every graph has stack-number bounded by a number, thus proving a conjecture of Heath, Leighton and function of its queue-number, or whether every graph has Rosenberg from 1992. The key to the proof is a new structural queue-number bounded by a function of its stack-number tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each [1,2]. part has a bounded number of vertices in each layer, and the Planar graphs are the simplest class of graphs where it is quotient graph has bounded treewidth. This result generalises unknown whether both stack and queue-number are bounded. for graphs of bounded Euler genus. Moreover, we prove that In particular, Buss and Shor [3] first proved that planar every graph in a minor-closed class has such a layered partition graphs have bounded stack-number; the best known upper if and only if the class excludes some apex graph. -
Packing and Covering Dense Graphs
Packing and Covering Dense Graphs Noga Alon ∗ Yair Caro y Raphael Yuster z Abstract Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G. Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants (H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 − (H)) G , then the H-packing number of G and the H-covering number of G can be computed j j in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G. Further extensions and solutions to related problems are also given. 1 Introduction All graphs considered here are finite, undirected and simple, unless otherwise noted. For the standard graph-theoretic terminology the reader is referred to [1]. Let H be a graph without isolated vertices. An H-covering of a graph G is a set L = G1;:::;Gs of subgraphs of G, where f g each subgraph is isomorphic to H, and every edge of G appears in at least one member of L. -
Embedding Trees in Dense Graphs
Embedding trees in dense graphs Václav Rozhoň February 14, 2019 joint work with T. Klimo¹ová and D. Piguet Václav Rozhoň Embedding trees in dense graphs February 14, 2019 1 / 11 Alternative definition: substructures in graphs Extremal graph theory Definition (Extremal graph theory, Bollobás 1976) Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians. Václav Rozhoň Embedding trees in dense graphs February 14, 2019 2 / 11 Extremal graph theory Definition (Extremal graph theory, Bollobás 1976) Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians. Alternative definition: substructures in graphs Václav Rozhoň Embedding trees in dense graphs February 14, 2019 2 / 11 Density of edges vs. density of triangles (Razborov 2008) (image from the book of Lovász) Extremal graph theory Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2=4 edges then it contains a triangle. Generalisations? Václav Rozhoň Embedding trees in dense graphs February 14, 2019 3 / 11 Extremal graph theory Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2=4 edges then it contains a triangle. Generalisations? Density of edges vs. density of triangles (Razborov 2008) (image from the book of Lovász) Václav Rozhoň Embedding trees in dense graphs February 14, 2019 3 / 11 3=2 The answer for C4 is of order n , lower bound via finite projective planes. What is the answer for trees? Extremal graph theory Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2=4 edges then it contains a triangle. -
Fractional Arboricity, Strength, and Principal Partitions in Graphs and Matroids
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 40 (1992) 285-302 285 North-Holland Fractional arboricity, strength, and principal partitions in graphs and matroids Paul A. Catlin Wayne State University, Detroit, MI 48202. USA Jerrold W. Grossman Oakland University, Rochester, MI 48309, USA Arthur M. Hobbs* Texas A & M University, College Station, TX 77843, USA Hong-Jian Lai** West Virginia University, Morgantown, WV 26506, USA Received 15 February 1989 Revised 15 November 1989 Abstract Catlin, P.A., J.W. Grossman, A.M. Hobbs and H.-J. Lai, Fractional arboricity, strength, and principal partitions in graphs and matroids, Discrete Applied Mathematics 40 (1992) 285-302. In a 1983 paper, D. Gusfield introduced a function which is called (following W.H. Cunningham, 1985) the strength of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link, this is the function v(G) = mmFr9c) IFl/(w(G F) - w(G)), where the minimum is taken over all subsets F of E(G) such that w(G F), the number of components of G - F, is at least w(G) + 1. In a 1986 paper, C. Payan introduced thefractional arboricity of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link this function is y(G) = maxHrC lE(H)I/(l V(H)1 -w(H)), where H runs over Correspondence to: Professor A.M. Hobbs, Department of Mathematics, Texas A & M University, College Station, TX 77843, USA. -
Limits of Graph Sequences
Snapshots of modern mathematics № 10/2019 from Oberwolfach Limits of graph sequences Tereza Klimošová Graphs are simple mathematical structures used to model a wide variety of real-life objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to ef- ficiently represent and study properties of extremely large graphs led to the development of the theory of graph limits. 1 Graphs A graph is one of the simplest mathematical structures. It consists of a set of vertices (which we depict as points) and edges between the pairs of vertices (depicted as line segments); several examples are shown in Figure 1. Many real-life settings can be modeled using graphs: for example, social and computer networks, road and other transport network maps, and the structure of molecules (see Figure 2a and 2b). These models are widely used in computer science, for instance for route planning algorithms or by Google’s PageRank algorithm, which generates search results. What these models all have in common is the representation of a set of several objects (the vertices) and relations or connections between pairs of those objects (the edges). In recent years, with the increasing use of computers in all areas of life, it has become necessary to handle large volumes of data. To represent such data (for example, connections between internet servers) in turn requires graphs with very many vertices and an even larger number of edges. In such situations, traditional algorithms for processing graphs are often impractical, or even impossible, because the computations take too much time and/or computational 1 Figure 1: Examples of graphs. -
Arxiv:2107.04993V1 [Cs.DS] 11 Jul 2021
The Mixed Page Number of Graphs Jawaherul Md. Alam1 , Michael A. Bekos2 , Martin Gronemann3 , Michael Kaufmann2 , and Sergey Pupyrev4 1 Amazon Inc., Tempe, AZ, USA [email protected] 2 Institut f¨urInformatik, Universit¨atT¨ubingen,T¨ubingen,Germany fbekos,[email protected] 3 Theoretical Computer Science, Osnabr¨uck University, Osnabr¨uck, Germany [email protected] 4 Facebook, Inc., Menlo Park, CA, USA [email protected] Abstract. A linear layout of a graph typically consists of a total vertex order, and a partition of the edges into sets of either non-crossing edges, called stacks, or non-nested edges, called queues. The stack (queue) number of a graph is the minimum number of required stacks (queues) in a linear layout. Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. In this work we initiate the study of the mixed page number of a graph which corresponds to the minimum number of such sets. First, we study the edge density of graphs with bounded mixed page number. Then, we focus on complete and complete bipartite graphs, for which we derive lower and upper bounds on their mixed page number. Our findings indicate that combining stacks and queues is more powerful in various ways compared to the two traditional layouts. Keywords: Linear layouts · Mixed page number · Stacks and queues 1 Introduction Linear layouts of graphs form an important research topic, which has been studied in different areas and under different perspectives. As a matter of fact, several combinatorial optimization problems are defined by means of a measure over a linear layout of a graph (including the well- known cutwidth [1], bandwidth [9] and pathwidth [23]). -
The $ K $-Strong Induced Arboricity of a Graph
The k-strong induced arboricity of a graph Maria Axenovich, Daniel Gon¸calves, Jonathan Rollin, Torsten Ueckerdt June 1, 2017 Abstract The induced arboricity of a graph G is the smallest number of induced forests covering the edges of G. This is a well-defined parameter bounded from above by the number of edges of G when each forest in a cover consists of exactly one edge. Not all edges of a graph necessarily belong to induced forests with larger components. For k > 1, we call an edge k-valid if it is contained in an induced tree on k edges. The k-strong induced arboricity of G, denoted by fk(G), is the smallest number of induced forests with components of sizes at least k that cover all k-valid edges in G. This parameter is highly non-monotone. However, we prove that for any proper minor-closed graph class C, and more gener- ally for any class of bounded expansion, and any k > 1, the maximum value of fk(G) for G 2 C is bounded from above by a constant depending only on C and k. This implies that the adjacent closed vertex-distinguishing number of graphs from a class of bounded expansion is bounded by a constant depending only on the class. We further t+1 d prove that f2(G) 6 3 3 for any graph G of tree-width t and that fk(G) 6 (2k) for any graph of tree-depth d. In addition, we prove that f2(G) 6 310 when G is planar. -
Layouts of Graph Subdivisions
Layouts of Graph Subdivisions Vida Dujmovi´c1,2 and David R. Wood2,3 1 School of Computer Science, McGill University, Montr´eal, Canada [email protected] 2 School of Computer Science, Carleton University, Ottawa, Canada [email protected] 3 Department of Applied Mathematics, Charles University, Prague, Czech Republic Abstract. A k-stack layout (respectively, k-queue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex or- dering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The stack-number (respectively, queue-number, track-number) of a graph G, denoted by sn(G)(qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout. This paper studies stack, queue, and track layouts of graph sub- divisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n)to O(log min{sn(G), qn(G)}). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number. It is proved that every graph has a 2- queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. -
Local Page Numbers
Local Page Numbers Bachelor Thesis of Laura Merker At the Department of Informatics Institute of Theoretical Informatics Reviewers: Prof. Dr. Dorothea Wagner Prof. Dr. Peter Sanders Advisor: Dr. Torsten Ueckerdt Time Period: May 22, 2018 – September 21, 2018 KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Statement of Authorship I hereby declare that this document has been composed by myself and describes my own work, unless otherwise acknowledged in the text. I declare that I have observed the Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis, as amended. Ich versichere wahrheitsgemäß, die Arbeit selbstständig verfasst, alle benutzten Hilfsmittel vollständig und genau angegeben und alles kenntlich gemacht zu haben, was aus Arbeiten anderer unverändert oder mit Abänderungen entnommen wurde, sowie die Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis in der jeweils gültigen Fassung beachtet zu haben. Karlsruhe, September 21, 2018 iii Abstract A k-local book embedding consists of a linear ordering of the vertices of a graph and a partition of its edges into sets of edges, called pages, such that any two edges on the same page do not cross and every vertex has incident edges on at most k pages. Here, two edges cross if their endpoints alternate in the linear ordering. The local page number pl(G) of a graph G is the minimum k such that there exists a k-local book embedding for G. Given a graph G and a vertex ordering, we prove that it is NP-complete to decide whether there exists a k-local book embedding for G with respect to the given vertex ordering for any fixed k ≥ 3. -
Sparse Exchangeable Graphs and Their Limits Via Graphon Processes
Journal of Machine Learning Research 18 (2018) 1{71 Submitted 8/16; Revised 6/17; Published 5/18 Sparse Exchangeable Graphs and Their Limits via Graphon Processes Christian Borgs [email protected] Microsoft Research One Memorial Drive Cambridge, MA 02142, USA Jennifer T. Chayes [email protected] Microsoft Research One Memorial Drive Cambridge, MA 02142, USA Henry Cohn [email protected] Microsoft Research One Memorial Drive Cambridge, MA 02142, USA Nina Holden [email protected] Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA Editor: Edoardo M. Airoldi Abstract In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in R+. Here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over σ-finite measure spaces, we can model a large family of exchangeable graphs, including the Caron-Fox graphs and the traditional exchangeable dense graphs as special cases. Explicitly, modelling the underlying space of features by a σ-finite measure space (S; S; µ) and the connection probabilities by an integrable function W : S × S ! [0; 1], we construct a random family (Gt)t≥0 of growing graphs such that the vertices of Gt are given by a Poisson point process on S with intensity tµ, with two points x; y of the point process connected with probability W (x; y). We call such a random family a graphon process. We prove that a graphon process has convergent subgraph frequencies (with possibly infinite limits) and that, in the natural extension of the cut metric to our setting, the sequence converges to the generating graphon. -
Sparse Graphs
EuroComb 2005 DMTCS proc. AE, 2005, 181–186 Pebble Game Algorithms and (k, l)-Sparse Graphs Audrey Lee1†and Ileana Streinu2‡ 1Department of Computer Science, University of Massachusetts at Amherst, Amherst, MA, USA. [email protected] 2Computer Science Department, Smith College, Northampton, MA, USA. [email protected] A multi-graph G on n vertices is (k, l)-sparse if every subset of n0 ≤ n vertices spans at most kn0 − l edges, 0 ≤ l < 2k. G is tight if, in addition, it has exactly kn − l edges. We characterize (k, l)-sparse graphs via a family of simple, elegant and efficient algorithms called the (k, l)-pebble games. As applications, we use the pebble games for computing components (maximal tight subgraphs) in sparse graphs, to obtain inductive (Henneberg) constructions, and, when l = k, edge-disjoint tree decompositions. Keywords: sparse graph, pebble game, rigidity, arboricity, graph orientation with bounded degree 1 Introduction A graph§ G = (V, E) with n = |V | vertices and m = |E| edges is (k, l)-sparse if every subset of n0 ≤ n vertices spans at most kn0 − l edges. If, furthermore, m = kn − l, G will be called tight. A graph is a (k, a)-arborescence if it can be decomposed into k edge-disjoint spanning trees after the addition of any a edges. Here, k, l and a are positive integers satisfying a simple relationship, which is 0 ≤ l < 2k for sparseness or 0 ≤ a < k for arboricity. Sparseness and arboricity are closely related, and have important applications in Rigidity Theory. Clas- sical results of Nash-Williams [10] and Tutte [16] identify the class of graphs decomposable into k edge- disjoint spanning trees with the tight (k, k)-sparse graphs.