P4-Free Partition and Cover Numbers and Application
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On the Tree–Depth of Random Graphs Arxiv:1104.2132V2 [Math.CO] 15 Feb 2012
On the tree–depth of random graphs ∗ G. Perarnau and O.Serra November 11, 2018 Abstract The tree–depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree–depth of random graphs. For dense graphs, p n−1, the tree–depth of a random graph G is a.a.s. td(G) = n − O(pn=p). Random graphs with p = c=n, have a.a.s. linear tree–depth when c > 1, the tree–depth is Θ(log n) when c = 1 and Θ(log log n) for c < 1. The result for c > 1 is derived from the computation of tree–width and provides a more direct proof of a conjecture by Gao on the linearity of tree–width recently proved by Lee, Lee and Oum [?]. We also show that, for c = 1, every width parameter is a.a.s. constant, and that random regular graphs have linear tree–depth. 1 Introduction An elimination tree of a graph G is a rooted tree on the set of vertices such that there are no edges in G between vertices in different branches of the tree. The natural elimination scheme provided by this tree is used in many graph algorithmic problems where two non adjacent subsets of vertices can be managed independently. One good example is the Cholesky decomposition of symmetric matrices (see [?, ?, ?, ?]). Given an elimination tree, a distributed algorithm can be designed which takes care of disjoint subsets of vertices in different parallel processors. Starting arXiv:1104.2132v2 [math.CO] 15 Feb 2012 by the furthest level from the root, it proceeds by exposing at each step the vertices at a given depth. -
Forbidding Subgraphs
Graph Theory and Additive Combinatorics Lecturer: Prof. Yufei Zhao 2 Forbidding subgraphs 2.1 Mantel’s theorem: forbidding a triangle We begin our discussion of extremal graph theory with the following basic question. Question 2.1. What is the maximum number of edges in an n-vertex graph that does not contain a triangle? Bipartite graphs are always triangle-free. A complete bipartite graph, where the vertex set is split equally into two parts (or differing by one vertex, in case n is odd), has n2/4 edges. Mantel’s theorem states that we cannot obtain a better bound: Theorem 2.2 (Mantel). Every triangle-free graph on n vertices has at W. Mantel, "Problem 28 (Solution by H. most bn2/4c edges. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff). Wiskundige Opgaven 10, 60 —61, 1907. We will give two proofs of Theorem 2.2. Proof 1. G = (V E) n m Let , a triangle-free graph with vertices and x edges. Observe that for distinct x, y 2 V such that xy 2 E, x and y N(x) must not share neighbors by triangle-freeness. Therefore, d(x) + d(y) ≤ n, which implies that d(x)2 = (d(x) + d(y)) ≤ mn. ∑ ∑ N(y) x2V xy2E y On the other hand, by the handshake lemma, ∑x2V d(x) = 2m. Now by the Cauchy–Schwarz inequality and the equation above, Adjacent vertices have disjoint neigh- borhoods in a triangle-free graph. !2 ! 4m2 = ∑ d(x) ≤ n ∑ d(x)2 ≤ mn2; x2V x2V hence m ≤ n2/4. -
The History of Degenerate (Bipartite) Extremal Graph Problems
The history of degenerate (bipartite) extremal graph problems Zolt´an F¨uredi and Mikl´os Simonovits May 15, 2013 Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary [email protected] and [email protected] Abstract This paper is a survey on Extremal Graph Theory, primarily fo- cusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions. 1 Contents 1 Introduction 4 1.1 Some central theorems of the field . 5 1.2 Thestructureofthispaper . 6 1.3 Extremalproblems ........................ 8 1.4 Other types of extremal graph problems . 10 1.5 Historicalremarks . .. .. .. .. .. .. 11 arXiv:1306.5167v2 [math.CO] 29 Jun 2013 2 The general theory, classification 12 2.1 The importance of the Degenerate Case . 14 2.2 The asymmetric case of Excluded Bipartite graphs . 15 2.3 Reductions:Hostgraphs. 16 2.4 Excluding complete bipartite graphs . 17 2.5 Probabilistic lower bound . 18 1 Research supported in part by the Hungarian National Science Foundation OTKA 104343, and by the European Research Council Advanced Investigators Grant 267195 (ZF) and by the Hungarian National Science Foundation OTKA 101536, and by the European Research Council Advanced Investigators Grant 321104. (MS). 1 F¨uredi-Simonovits: Degenerate (bipartite) extremal graph problems 2 2.6 Classification of extremal problems . 21 2.7 General conjectures on bipartite graphs . 23 3 Excluding complete bipartite graphs 24 3.1 Bipartite C4-free graphs and the Zarankiewicz problem . 24 3.2 Finite Geometries and the C4-freegraphs . -
Finite Geometries
Finite Geometries Gy¨orgyKiss June 26th, 2012, Rogla gyk Finite Geometries Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: The distance of two elements d(x; y) : length of the shortest chain joining them. Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. gyk Finite Geometries The distance of two elements d(x; y) : length of the shortest chain joining them. Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: gyk Finite Geometries Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: The distance of two elements d(x; y) : length of the shortest chain joining them. gyk Finite Geometries Generalized polygons Definition Let n > 1 be a positive integer. -
Testing Isomorphism in the Bounded-Degree Graph Model (Preliminary Version)
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 102 (2019) Testing Isomorphism in the Bounded-Degree Graph Model (preliminary version) Oded Goldreich∗ August 11, 2019 Abstract We consider two versions of the problem of testing graph isomorphism in the bounded-degree graph model: A version in which one graph is fixed, and a version in which the input consists of two graphs. We essentially determine the query complexity of these testing problems in the special case of n-vertex graphs with connected components of size at most poly(log n). This is done by showing that these problems are computationally equivalent (up to polylogarithmic factors) to corresponding problems regarding isomorphism between sequences (over a large alphabet). Ignoring the dependence on the proximity parameter, our main results are: 1. The query complexity of testing isomorphism to a fixed object (i.e., an n-vertex graph or an n-long sequence) is Θ(e n1=2). 2. The query complexity of testing isomorphism between two input objects is Θ(e n2=3). Testing isomorphism between two sequences is shown to be related to testing that two distributions are equivalent, and this relation yields reductions in three of the four relevant cases. Failing to reduce the problem of testing the equivalence of two distribution to the problem of testing isomorphism between two sequences, we adapt the proof of the lower bound on the complexity of the first problem to the second problem. This adaptation constitutes the main technical contribution of the current work. Determining the complexity of testing graph isomorphism (in the bounded-degree graph model), in the general case (i.e., for arbitrary bounded-degree graphs), is left open. -
Practical Verification of MSO Properties of Graphs of Bounded Clique-Width
Practical verification of MSO properties of graphs of bounded clique-width Ir`ene Durand (joint work with Bruno Courcelle) LaBRI, Universit´ede Bordeaux 20 Octobre 2010 D´ecompositions de graphes, th´eorie, algorithmes et logiques, 2010 Objectives Verify properties of graphs of bounded clique-width Properties ◮ connectedness, ◮ k-colorability, ◮ existence of cycles ◮ existence of paths ◮ bounds (cardinality, degree, . ) ◮ . How : using term automata Note that we consider finite graphs only 2/33 Graphs as relational structures For simplicity, we consider simple,loop-free undirected graphs Extensions are easy Every graph G can be identified with the relational structure (VG , edgG ) where VG is the set of vertices and edgG ⊆VG ×VG the binary symmetric relation that defines edges. v7 v6 v2 v8 v1 v3 v5 v4 VG = {v1, v2, v3, v4, v5, v6, v7, v8} edgG = {(v1, v2), (v1, v4), (v1, v5), (v1, v7), (v2, v3), (v2, v6), (v3, v4), (v4, v5), (v5, v8), (v6, v7), (v7, v8)} 3/33 Expression of graph properties First order logic (FO) : ◮ quantification on single vertices x, y . only ◮ too weak ; can only express ”local” properties ◮ k-colorability (k > 1) cannot be expressed Second order logic (SO) ◮ quantifications on relations of arbitrary arity ◮ SO can express most properties of interest in Graph Theory ◮ too complex (many problems are undecidable or do not have a polynomial solution). 4/33 Monadic second order logic (MSO) ◮ SO formulas that only use quantifications on unary relations (i.e., on sets). ◮ can express many useful graph properties like connectedness, k-colorability, planarity... Example : k-colorability Stable(X ) : ∀u, v(u ∈ X ∧ v ∈ X ⇒¬edg(u, v)) Partition(X1,..., Xm) : ∀x(x ∈ X1 ∨ . -
Harary Polynomials
numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 1:2 (2021) Article #S2R13 ecajournal.haifa.ac.il Harary Polynomials Orli Herscovici∗, Johann A. Makowskyz and Vsevolod Rakitay ∗Department of Mathematics, University of Haifa, Israel Email: [email protected] zDepartment of Computer Science, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] yDepartment of Mathematics, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] Received: November 13, 2020, Accepted: February 7, 2021, Published: February 19, 2021 The authors: Released under the CC BY-ND license (International 4.0) Abstract: Given a graph property P, F. Harary introduced in 1985 P-colorings, graph colorings where each color class induces a graph in P. Let χP (G; k) counts the number of P-colorings of G with at most k colors. It turns out that χP (G; k) is a polynomial in Z[k] for each graph G. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial χ(G; k). We show that the characteristic and the Laplacian polynomial, the matching, the independence and the domination polynomials are not Harary polynomials. We show that for various notions of sparse, non-trivial properties P, the polynomial χP (G; k) is, in contrast to χ(G; k), not a chromatic, and even not an edge elimination invariant. Finally, we study whether the Harary polynomials are definable in monadic second-order Logic. Keywords: Generalized colorings; Graph polynomials; Courcelle's Theorem 2020 Mathematics Subject Classification: 05; 05C30; 05C31 1. -
Every Monotone Graph Property Is Testable∗
Every Monotone Graph Property is Testable∗ Noga Alon † Asaf Shapira ‡ Abstract A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on . This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemer´edi’s Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P, any graph that is -far from satisfying P, contains a subgraph of size depending on only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is -far from satisfying a (possibly infinite) set of monotone graph properties P, then it is at least δP ()-far from satisfying one of the properties. -
Graph Properties and Hypergraph Colourings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 98 (1991) 81-93 81 North-Holland Graph properties and hypergraph colourings Jason I. Brown Department of Mathematics, York University, 4700 Keele Street, Toronto, Ont., Canada M3l lP3 Derek G. Corneil Department of Computer Science, University of Toronto, Toronto, Ont., Canada MSS IA4 Received 20 December 1988 Revised 13 June 1990 Abstract Brown, J.I., D.G. Corneil, Graph properties and hypergraph colourings, Discrete Mathematics 98 (1991) 81-93. Given a graph property P, graph G and integer k 20, a P k-colouring of G is a function Jr:V(G)+ (1,. ) k} such that the subgraph induced by each colour class has property P. When P is closed under induced subgraphs, we can construct a hypergraph HG such that G is P k-colourable iff Hg is k-colourable. This correlation enables us to derive interesting new results in hypergraph chromatic theory from a ‘graphic’ approach. In particular, we build vertex critical hypergraphs that are not edge critical, construct uniquely colourable hypergraphs with few edges and find graph-to-hypergraph transformations that preserve chromatic numbers. 1. Introduction A property P is a collection of graphs (closed under isomorphism) containing K, and K, (all graphs considered here are finite); P is hereditary iff P is closed under induced subgraphs and nontrivial iff P does not contain every graph. Any graph G E P is called a P-graph. Throughout, we shall only be interested in hereditary and nontrivial properties, and P will always denote such a property. -
Nearly Tight Approximability Results for Minimum Biclique Cover and Partition
Nearly Tight Approximability Results for Minimum Biclique Cover and Partition Parinya Chalermsook1, Sandy Heydrich1;3, Eugenia Holm2, and Andreas Karrenbauer1∗ 1 Max Planck Institute for Informatics, Saarbr¨ucken, Germany 2 Dept. for Computer and Information Science, University of Konstanz, Germany 3 Saarbr¨ucken Graduate School of Computer Science {andreas.karrenbauer,parinya.chalermsook,sandy.heydrich}@mpi-inf.mpg.de [email protected] Abstract. In this paper, we consider the minimum biclique cover and minimum biclique partition problems on bipartite graphs. In the min- imum biclique cover problem, we are given an input bipartite graph G = (V; E), and our goal is to compute the minimum number of complete bipartite subgraphs that cover all edges of G. This problem, besides its correspondence to a well-studied notion of bipartite dimension in graph theory, has applications in many other research areas such as artificial intelligence, computer security, automata theory, and biology. Since it is NP-hard, past research has focused on approximation algorithms, fixed parameter tractability, and special graph classes that admit polynomial time exact algorithms. For the minimum biclique partition problem, we are interested in a biclique cover that covers each edge exactly once. We revisit the problems from approximation algorithms' perspectives and give nearly tight lower and upper bound results. We first show that both problems are NP-hard to approximate to within a factor of n1−" (where n is the number of vertices in the input graph). Using a stronger complexity assumption, the hardness becomes Ω~(n), where Ω~(·) hides lower order terms. Then we show that approximation factors of the form n=(log n)γ for some γ > 0 can be obtained. -
Isomorphism and Embedding Problems for Infinite Limits of Scale
Isomorphism and Embedding Problems for Infinite Limits of Scale-Free Graphs Robert D. Kleinberg ∗ Jon M. Kleinberg y Abstract structure of finite PA graphs; in particular, we give a The study of random graphs has traditionally been characterization of the graphs H for which the expected dominated by the closely-related models (n; m), in number of subgraph embeddings of H in an n-node PA which a graph is sampled from the uniform distributionG graph remains bounded as n goes to infinity. on graphs with n vertices and m edges, and (n; p), in n G 1 Introduction which each of the 2 edges is sampled independently with probability p. Recen tly, however, there has been For decades, the study of random graphs has been dom- considerable interest in alternate random graph models inated by the closely-related models (n; m), in which designed to more closely approximate the properties of a graph is sampled from the uniformG distribution on complex real-world networks such as the Web graph, graphs with n vertices and m edges, and (n; p), in n G the Internet, and large social networks. Two of the most which each of the 2 edges is sampled independently well-studied of these are the closely related \preferential with probability p.The first was introduced by Erd}os attachment" and \copying" models, in which vertices and R´enyi in [16], the second by Gilbert in [19]. While arrive one-by-one in sequence and attach at random in these random graphs have remained a central object \rich-get-richer" fashion to d earlier vertices. -
Covering Graphs with Few Complete Bipartite Subgraphs *
Covering Graphs with Few Complete Bipartite Subgraphs ⋆ Herbert Fleischner1, Egbert Mujuni2,⋆⋆, Dani¨el Paulusma3⋆ ⋆ ⋆, and Stefan Szeider3† 1 Department of Computer Science, Vienna Technical University A-1040 Vienna, Austria [email protected] 2 Mathematics Department, University of Dar es Salaam PO Box 35062, Dar es Salaam, Tanzania [email protected] 3 Department of Computer Science, Durham University Durham DH1 3LE, United Kingdom {daniel.paulusma,stefan.szeider}@durham.ac.uk Abstract. We consider computational problems on covering graphs with bicliques (complete bipartite subgraphs). Given a graph and an inte- ger k, the biclique cover problem asks whether the edge-set of the graph can be covered with at most k bicliques; the biclique partition problem is defined similarly with the additional condition that the bicliques are required to be mutually edge-disjoint. The biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most k bicliques, the biclique vertex-partition problem is defined simi- larly with the additional condition that the bicliques are required to be mutually vertex-disjoint. All these four problems are known to be NP- complete even if the given graph is bipartite. In this paper we investigate them in the framework of parameterized complexity: do the problems become easier if k is assumed to be small? We show that, considering k as the parameter, the first two problems are fixed-parameter tractable, while the latter two problems are not fixed-parameter tractable unless P = NP. Keywords. bicliques, parameterized complexity, covering and parti- tioning problems. 1 Introduction Let G be a simple undirected graph and let S be a set of (not necessarily vertex- induced) subgraphs of G.