On the Thinness and Proper Thinness of a Graph
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Lecture 10: April 20, 2005 Perfect Graphs
Re-revised notes 4-22-2005 10pm CMSC 27400-1/37200-1 Combinatorics and Probability Spring 2005 Lecture 10: April 20, 2005 Instructor: L´aszl´oBabai Scribe: Raghav Kulkarni TA SCHEDULE: TA sessions are held in Ryerson-255, Monday, Tuesday and Thursday 5:30{6:30pm. INSTRUCTOR'S EMAIL: [email protected] TA's EMAIL: [email protected], [email protected] IMPORTANT: Take-home test Friday, April 29, due Monday, May 2, before class. Perfect Graphs k 1=k Shannon capacity of a graph G is: Θ(G) := limk (α(G )) : !1 Exercise 10.1 Show that α(G) χ(G): (G is the complement of G:) ≤ Exercise 10.2 Show that χ(G H) χ(G)χ(H): · ≤ Exercise 10.3 Show that Θ(G) χ(G): ≤ So, α(G) Θ(G) χ(G): ≤ ≤ Definition: G is perfect if for all induced sugraphs H of G, α(H) = χ(H); i. e., the chromatic number is equal to the clique number. Theorem 10.4 (Lov´asz) G is perfect iff G is perfect. (This was open under the name \weak perfect graph conjecture.") Corollary 10.5 If G is perfect then Θ(G) = α(G) = χ(G): Exercise 10.6 (a) Kn is perfect. (b) All bipartite graphs are perfect. Exercise 10.7 Prove: If G is bipartite then G is perfect. Do not use Lov´asz'sTheorem (Theorem 10.4). 1 Lecture 10: April 20, 2005 2 The smallest imperfect (not perfect) graph is C5 : α(C5) = 2; χ(C5) = 3: For k 2, C2k+1 imperfect. -
I?'!'-Comparability Graphs
Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors). -
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. -
Box Representations of Embedded Graphs
Box representations of embedded graphs Louis Esperet CNRS, Laboratoire G-SCOP, Grenoble, France S´eminairede G´eom´etrieAlgorithmique et Combinatoire, Paris March 2017 Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. -
Improved Ramsey-Type Results in Comparability Graphs
Improved Ramsey-type results in comparability graphs D´anielKor´andi ∗ Istv´anTomon ∗ Abstract Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size n1=(r+1). This bound is known to be tight for r = 1. The question whether it is optimal for r ≥ 2 was studied by Dumitrescu and T´oth.We prove that it is essentially best possible for r = 2, as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size n1=3+o(1). Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G, nor its complement contains a complete bipartite graph with parts of size cn (log n)r . With this, we improve a result of Fox and Pach. 1 Introduction An old problem of Larman, Matouˇsek,Pach and T¨or}ocsik[11, 14] asks for the largest m = m(n) such that among any n convex sets in the plane, there are m that are pairwise disjoint or m that pairwise intersect. Considering the disjointness graph G, whose vertices correspond to the convex sets and edges correspond to disjoint pairs, this problem asks the Ramsey-type question of estimating the largest clique or independent set in G. -
P 4-Colorings and P 4-Bipartite Graphs Chinh T
P_4-Colorings and P_4-Bipartite Graphs Chinh T. Hoàng, van Bang Le To cite this version: Chinh T. Hoàng, van Bang Le. P_4-Colorings and P_4-Bipartite Graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, 4 (2), pp.109-122. hal-00958951 HAL Id: hal-00958951 https://hal.inria.fr/hal-00958951 Submitted on 13 Mar 2014 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. Discrete Mathematics and Theoretical Computer Science 4, 2001, 109–122 P4-Free Colorings and P4-Bipartite Graphs Ch´ınh T. Hoang` 1† and Van Bang Le2‡ 1Department of Physics and Computing, Wilfrid Laurier University, 75 University Ave. W., Waterloo, Ontario N2L 3C5, Canada 2Fachbereich Informatik, Universitat¨ Rostock, Albert-Einstein-Straße 21, D-18051 Rostock, Germany received May 19, 1999, revised November 25, 2000, accepted December 15, 2000. A vertex partition of a graph into disjoint subsets Vis is said to be a P4-free coloring if each color class Vi induces a subgraph without a chordless path on four vertices (denoted by P4). Examples of P4-free 2-colorable graphs (also called P4-bipartite graphs) include parity graphs and graphs with “few” P4s like P4-reducible and P4-sparse graphs. -
On Sources in Comparability Graphs, with Applications Stephan Olariu Old Dominion University
Old Dominion University ODU Digital Commons Computer Science Faculty Publications Computer Science 1992 On Sources in Comparability Graphs, With Applications Stephan Olariu Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/computerscience_fac_pubs Part of the Applied Mathematics Commons, and the Computer Sciences Commons Repository Citation Olariu, Stephan, "On Sources in Comparability Graphs, With Applications" (1992). Computer Science Faculty Publications. 128. https://digitalcommons.odu.edu/computerscience_fac_pubs/128 Original Publication Citation Olariu, S. (1992). On sources in comparability graphs, with applications. Discrete Mathematics, 110(1-3), 289-292. doi:10.1016/ 0012-365x(92)90721-q This Article is brought to you for free and open access by the Computer Science at ODU Digital Commons. It has been accepted for inclusion in Computer Science Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Discrete Mathematics 110 (1992) 2X9-292 289 North-Holland Note On sources in comparability graphs, with applications S. Olariu Department of Computer Science, Old Dominion University, Norfolk, VA 235294162, USA Received 31 October 1989 Abstract Olariu, S., On sources in comparability graphs, with applications, Discrete Mathematics 110 (1992) 289-292. We characterize sources in comparability graphs and show that our result provides a unifying look at two recent results about interval graphs. An orientation 0 of a graph G is obtained by assigning unique directions to its edges. To simplify notation, we write xy E 0, whenever the edge xy receives the direction from x to y. A vertex w is called a source whenever VW E 0 for no vertex II in G. -
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models Von der Fakult¨at fur¨ Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker George B. Mertzios aus Thessaloniki, Griechenland Berichter: Privat Dozent Dr. Walter Unger (Betreuer) Professor Dr. Berthold V¨ocking (Zweitbetreuer) Professor Dr. Dieter Rautenbach Tag der mundlichen¨ Prufung:¨ Montag, den 30. November 2009 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ Abstract This thesis mainly deals with the structure of some classes of perfect graphs that have been widely investigated, due to both their interesting structure and their numerous applications. By exploiting the structure of these graph classes, we provide solutions to some open problems on them (in both the affirmative and negative), along with some new representation models that enable the design of new efficient algorithms. In particular, we first investigate the classes of interval and proper interval graphs, and especially, path problems on them. These classes of graphs have been extensively studied and they find many applications in several fields and disciplines such as genetics, molecular biology, scheduling, VLSI design, archaeology, and psychology, among others. Although the Hamiltonian path problem is well known to be linearly solvable on interval graphs, the complexity status of the longest path problem, which is the most natural optimization version of the Hamiltonian path problem, was an open question. We present the first polynomial algorithm for this problem with running time O(n4). Furthermore, we introduce a matrix representation for both interval and proper interval graphs, called the Normal Interval Representation (NIR) and the Stair Normal Interval Representation (SNIR) matrix, respectively. -
The Micro-World of Cographs
The Micro-world of Cographs Bogdan Alecu Vadim Lozin Dominique de Werra June 10, 2020 B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 1 / 26 Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S. A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class. We say p is bounded in X if there is a constant k such that p(G) ≤ k for all G 2 X , and unbounded in X otherwise. We work with finite, simple, undirected graphs. A hereditary class (just \class" from now on) is a set of graphs closed under taking induced subgraphs. Basic definitions B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S. A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class.