Cograph-(K, ) Graph Sandwich Problem
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Symmetry and Structure of Graphs
Symmetry and Structure of graphs by Kovács Máté Submitted to Central European University Department of Department of Mathematics and its Applications In partial fulfillment of the requirements for the degree of Master of Science Supervisor: Dr. Hegedus˝ Pál CEU eTD Collection Budapest, Hungary 2014 I, the undersigned [Kovács Máté], candidate for the degree of Master of Science at the Central European University Department of Mathematics and its Applications, declare herewith that the present thesis is exclusively my own work, based on my research and only such external information as properly credited in notes and bibliography. I declare that no unidentified and illegitimate use was made of work of others, and no part the thesis infringes on any person’s or institution’s copyright. I also declare that no part the thesis has been submitted in this form to any other institution of higher education for an academic degree. Budapest, 9 May 2014 ————————————————— Signature CEU eTD Collection c by Kovács Máté, 2014 All Rights Reserved. ii Abstract The thesis surveys results on structure and symmetry of graphs. Structure and symmetry of graphs can be handled by graph homomorphisms and graph automorphisms - the two approaches are compatible. Two graphs are called homomorphically equivalent if there is a graph homomorphism between the two graphs back and forth. Being homomorphically equivalent is an equivalence relation, and every class has a vertex minimal element called the graph core. It turns out that transitive graphs have transitive cores. The possibility of a structural result regarding transitive graphs is investigated. We speculate that almost all transitive graphs are cores. -
Induced Minors and Well-Quasi-Ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck
Induced minors and well-quasi-ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck To cite this version: Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck. Induced minors and well-quasi-ordering. EuroComb: European Conference on Combinatorics, Graph Theory and Appli- cations, Aug 2015, Bergen, Norway. pp.197-201, 10.1016/j.endm.2015.06.029. lirmm-01349277 HAL Id: lirmm-01349277 https://hal-lirmm.ccsd.cnrs.fr/lirmm-01349277 Submitted on 27 Jul 2016 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. Available online at www.sciencedirect.com www.elsevier.com/locate/endm Induced minors and well-quasi-ordering Jaroslaw Blasiok 1,2 School of Engineering and Applied Sciences, Harvard University, United States Marcin Kami´nski 2 Institute of Computer Science, University of Warsaw, Poland Jean-Florent Raymond 2 Institute of Computer Science, University of Warsaw, Poland, and LIRMM, University of Montpellier, France Th´eophile Trunck 2 LIP, ENS´ de Lyon, France Abstract AgraphH is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor- free. -
Arxiv:1907.08517V3 [Math.CO] 12 Mar 2021
RANDOM COGRAPHS: BROWNIAN GRAPHON LIMIT AND ASYMPTOTIC DEGREE DISTRIBUTION FRÉDÉRIQUE BASSINO, MATHILDE BOUVEL, VALENTIN FÉRAY, LUCAS GERIN, MICKAËL MAAZOUN, AND ADELINE PIERROT ABSTRACT. We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0; 1]. We finally an- alyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combi- natorial arguments, including the symbolic method and singularity analysis. 1. INTRODUCTION 1.1. Motivation. Random graphs are arguably the most studied objects at the interface of com- binatorics and probability theory. One aspect of their study consists in analyzing a uniform random graph of large size n in a prescribed family, e.g. perfect graphs [MY19], planar graphs [Noy14], graphs embeddable in a surface of given genus [DKS17], graphs in subcritical classes [PSW16], hereditary classes [HJS18] or addable classes [MSW06, CP19]. The present paper focuses on uniform random cographs (both in the labeled and unlabeled settings). Cographs were introduced in the seventies by several authors independently, see e.g. -
Lecture 12: Matroids 12.1 Matroids: Definition and Examples
IE 511: Integer Programming, Spring 2021 4 Mar, 2021 Lecture 12: Matroids Lecturer: Karthik Chandrasekaran Scribe: Karthik Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publi- cations. Matroids are structures that can be used to model the feasible set of several combinatorial opti- mization problems. In this lecture, we will define matroids, consider some examples, introduce the matroid optimization problem and see its generality through some concrete optimization problems that it formulates, and see a BIP formulation of the matroid optimization problem. We will sub- sequently see that the LP-relaxation of the BIP formulation has an integral optimal solution (via the concept of TDI that we learnt in the previous lecture). This will ultimately broaden the family of IPs that can be solved by simply solving the LP relaxation. 12.1 Matroids: Definition and Examples Definition 1. Let N be a finite set and I ⊆ 2N (recall that 2N denotes the collection of all subsets of N, i.e., 2N := fS : S ⊆ Ng). 1. The pair (N; I) is an independence system if (i) ; 2 I and (ii) If A 2 I and B ⊆ A, then B 2 I (i.e., the set I has the hereditary property{see Figure 12.1). The sets in I are called independent sets. Figure 12.1: Hereditary property of independent sets: If A is an independent set and B is a subset of A, then B should also be an independent set. 2. An independence system (N; I) is a matroid if (iii) for all A; B 2 I with jBj > jAj there exists e 2 B n A such that A [ feg 2 I. -
Finding Articulation Points and Bridges Articulation Points Articulation Point
Finding Articulation Points and Bridges Articulation Points Articulation Point Articulation Point A vertex v is an articulation point (also called cut vertex) if removing v increases the number of connected components. A graph with two articulation points. 3 / 1 Articulation Points Given I An undirected, connected graph G = (V; E) I A DFS-tree T with the root r Lemma A DFS on an undirected graph does not produce any cross edges. Conclusion I If a descendant u of a vertex v is adjacent to a vertex w, then w is a descendant or ancestor of v. 4 / 1 Removing a Vertex v Assume, we remove a vertex v 6= r from the graph. Case 1: v is an articulation point. I There is a descendant u of v which is no longer reachable from r. I Thus, there is no edge from the tree containing u to the tree containing r. Case 2: v is not an articulation point. I All descendants of v are still reachable from r. I Thus, for each descendant u, there is an edge connecting the tree containing u with the tree containing r. 5 / 1 Removing a Vertex v Problem I v might have multiple subtrees, some adjacent to ancestors of v, and some not adjacent. Observation I A subtree is not split further (we only remove v). Theorem A vertex v is articulation point if and only if v has a child u such that neither u nor any of u's descendants are adjacent to an ancestor of v. Question I How do we determine this efficiently for all vertices? 6 / 1 Detecting Descendant-Ancestor Adjacency Lowpoint The lowpoint low(v) of a vertex v is the lowest depth of a vertex which is adjacent to v or a descendant of v. -
Universality for and in Induced-Hereditary Graph Properties
Discussiones Mathematicae Graph Theory 33 (2013) 33–47 doi:10.7151/dmgt.1671 Dedicated to Mieczys law Borowiecki on his 70th birthday UNIVERSALITY FOR AND IN INDUCED-HEREDITARY GRAPH PROPERTIES Izak Broere Department of Mathematics and Applied Mathematics University of Pretoria e-mail: [email protected] and Johannes Heidema Department of Mathematical Sciences University of South Africa e-mail: [email protected] Abstract The well-known Rado graph R is universal in the set of all countable graphs , since every countable graph is an induced subgraph of R. We I study universality in and, using R, show the existence of 2ℵ0 pairwise non-isomorphic graphsI which are universal in and denumerably many other universal graphs in with prescribed attributes.I Then we contrast universality for and universalityI in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2ℵ0 ) properties in the lattice K of induced-hereditary properties of which ≤ only at most 2ℵ0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property. Keywords: countable graph, universal graph, induced-hereditary property. 2010 Mathematics Subject Classification: 05C63. 34 I.Broere and J. Heidema 1. Introduction and Motivation In this article a graph shall (with one illustrative exception) be simple, undirected, unlabelled, with a countable (i.e., finite or denumerably infinite) vertex set. For graph theoretical notions undefined here, we generally follow [14]. -
Vertex Deletion Problems on Chordal Graphs∗†
Vertex Deletion Problems on Chordal Graphs∗† Yixin Cao1, Yuping Ke2, Yota Otachi3, and Jie You4 1 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 2 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 3 Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, Japan [email protected] 4 School of Information Science and Engineering, Central South University and Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] Abstract Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yan- nakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. 1998 ACM Subject Classification F.2.2 Analysis of Algorithms and Problem Complexity, G.2.2 Graph Theory Keywords and phrases vertex deletion problem, maximum subgraph, chordal graph, (unit) in- terval graph, split graph, hereditary property, NP-complete, polynomial-time algorithm Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2017.22 1 Introduction Generally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices. -
A Comparison of Two Approaches for Polynomial Time Algorithms
A comparison of two approaches for polynomial time algorithms computing basic graph parameters∗† Frank Gurski‡ March 26, 2018 Abstract In this paper we compare and illustrate the algorithmic use of graphs of bounded tree- width and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-width in polynomial time. We also present linear time algorithms for computing the latter four basic graph parameters on trees, i.e. graphs of tree-width 1, and on co-graphs, i.e. graphs of clique-width at most 2. Keywords:graph algorithms, graph parameters, clique-width, NLC-width, tree-width 1 Introduction A graph parameter is a mapping that associates every graph with a positive integer. Well known graph parameters are independence number, dominating number, and chromatic num- ber. In general the computation of such parameters for some given graph is NP-hard. In this work we give fixed-parameter tractable (fpt) algorithms for computing basic graph parameters restricted to graph classes of bounded tree-width and graph classes of bounded clique-width. The tree-width of graphs has been defined in 1976 by Halin [Hal76] and independently in arXiv:0806.4073v1 [cs.DS] 25 Jun 2008 1986 by Robertson and Seymour [RS86] by the existence of a tree decomposition. Intuitively, the tree-width of some graph G measures how far G differs from a tree. Two more powerful and more recent graph parameters are clique-width1 and NLC-width2 both defined in 1994, by Courcelle and Olariu [CO00] and by Wanke [Wan94], respectively. -
The Strong Perfect Graph Theorem
Annals of Mathematics, 164 (2006), 51–229 The strong perfect graph theorem ∗ ∗ By Maria Chudnovsky, Neil Robertson, Paul Seymour, * ∗∗∗ and Robin Thomas Abstract A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu´ejols and Vuˇskovi´c — that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge’s conjecture cannot have either of these properties). In this paper we prove both of these conjectures. 1. Introduction We begin with definitions of some of our terms which may be nonstandard. All graphs in this paper are finite and simple. The complement G of a graph G has the same vertex set as G, and distinct vertices u, v are adjacent in G just when they are not adjacent in G.Ahole of G is an induced subgraph of G which is a cycle of length at least 4. An antihole of G is an induced subgraph of G whose complement is a hole in G. A graph G is Berge if every hole and antihole of G has even length. A clique in G is a subset X of V (G) such that every two members of X are adjacent. -
Algorithms for Deletion Problems on Split Graphs
Algorithms for deletion problems on split graphs Dekel Tsur∗ Abstract In the Split to Block Vertex Deletion and Split to Threshold Vertex Deletion problems the input is a split graph G and an integer k, and the goal is to decide whether there is a set S of at most k vertices such that G − S is a block graph and G − S is a threshold graph, respectively. In this paper we give algorithms for these problems whose running times are O∗(2.076k) and O∗(2.733k), respectively. Keywords graph algorithms, parameterized complexity. 1 Introduction A graph G is called a split graph if its vertex set can be partitioned into two disjoint sets C and I such that C is a clique and I is an independent set. A graph G is a block graph if every biconnected component of G is a clique. A graph G is a threshold graph if there is a t ∈ R and a function f : V (G) → R such that for every u, v ∈ V (G), (u, v) is an edge in G if and only if f(u)+ f(v) ≥ t. In the Split to Block Vertex Deletion (SBVD) problem the input is a split graph G and an integer k, and the goal is to decide whether there is a set S of at most k vertices such that G − S is a block graph. Similarly, in the Split to Threshold Vertex Deletion (STVD) problem the input is a split graph G and an integer k, and the goal is to decide whether there is a set S of at most k vertices such that G − S is a threshold graph. -
Partitioning a Graph Into Disjoint Cliques and a Triangle-Free Graph
This is a repository copy of Partitioning a graph into disjoint cliques and a triangle-free graph. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/85292/ Version: Accepted Version Article: Abu-Khzam, FN, Feghali, C and Muller, H (2015) Partitioning a graph into disjoint cliques and a triangle-free graph. Discrete Applied Mathematics, 190-19. 1 - 12. ISSN 0166-218X https://doi.org/10.1016/j.dam.2015.03.015 © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph Faisal N. Abu-Khzam, Carl Feghali, Haiko M¨uller Abstract A graph G =(V, E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques and B induces a triangle- free graph. -
Arxiv:1906.05510V2 [Math.AC]
BINOMIAL EDGE IDEALS OF COGRAPHS THOMAS KAHLE AND JONAS KRUSEMANN¨ Abstract. We determine the Castelnuovo–Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with n vertices the maximum regularity grows as 2n/3. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda. 1. Introduction Let G = ([n], E) be a simple undirected graph on the vertex set [n]= {1,...,n}. x1 ··· xn x1 ··· xn Let X = ( y1 ··· yn ) be a generic 2 × n matrix and S = k[ y1 ··· yn ] the polynomial ring whose indeterminates are the entries of X and with coefficients in a field k. The binomial edge ideal of G is JG = hxiyj −yixj : {i, j} ∈ Ei⊆ S, the ideal of 2×2 mi- nors indexed by the edges of the graph. Since their inception in [5, 15], connecting combinatorial properties of G with algebraic properties of JG or S/ JG has been a popular activity. Particular attention has been paid to the minimal free resolution of S/ JG as a standard N-graded S-module [3, 11]. The data of a minimal free res- olution is encoded in its graded Betti numbers βi,j (S/ JG) = dimk Tori(S/ JG, k)j . An interesting invariant is the highest degree appearing in the resolution, the Castelnuovo–Mumford regularity reg(S/ JG) = max{j − i : βij (S/ JG) 6= 0}. It is a complexity measure as low regularity implies favorable properties like vanish- ing of local cohomology. Binomial edge ideals have square-free initial ideals by arXiv:1906.05510v2 [math.AC] 10 Mar 2021 [5, Theorem 2.1] and, using [1], this implies that the extremal Betti numbers and regularity can also be derived from those initial ideals.