Characterizations of Cographs As Intersection Graphs of Paths on a Grid
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Forbidden Subgraph Characterization of Quasi-Line Graphs Medha Dhurandhar [email protected]
Forbidden Subgraph Characterization of Quasi-line Graphs Medha Dhurandhar [email protected] Abstract: Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques. 1. Introduction: A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. In [2] Chudnovsky and Seymour gave a constructive characterization of quasi-line graphs. An alternative characterization of quasi-line graphs is given in [3] stating that a graph has a fuzzy reconstruction iff it is a quasi-line graph and also in [4] using the concept of sums of Hoffman graphs. Here we characterize quasi-line graphs in terms of the forbidden induced subgraphs like line graphs. We consider in this paper only finite, simple, connected, undirected graphs. The vertex set of G is denoted by V(G), the edge set by E(G), the maximum degree of vertices in G by Δ(G), the maximum clique size by (G) and the chromatic number by G). N(u) denotes the neighbourhood of u and N(u) = N(u) + u. For further notation please refer to Harary [3]. 2. Main Result: Before proving the main result we prove some lemmas, which will be used later. Lemma 1: If G is {3K1, C5}-free, then either 1) G ~ K|V(G)| or 2) If v, w V(G) are s.t. -
Infinitely Many Minimal Classes of Graphs of Unbounded Clique-Width∗
Infinitely many minimal classes of graphs of unbounded clique-width∗ A. Collins, J. Foniok†, N. Korpelainen, V. Lozin, V. Zamaraev Abstract The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width. Keywords: clique-width, linear clique-width, hereditary class 1 Introduction Clique-width is a graph parameter which is important in theoretical computer science, because many algorithmic problems that are generally NP-hard become polynomial-time solvable when restricted to graphs of bounded clique-width [4]. -
Networkx: Network Analysis with Python
NetworkX: Network Analysis with Python Salvatore Scellato Full tutorial presented at the XXX SunBelt Conference “NetworkX introduction: Hacking social networks using the Python programming language” by Aric Hagberg & Drew Conway Outline 1. Introduction to NetworkX 2. Getting started with Python and NetworkX 3. Basic network analysis 4. Writing your own code 5. You are ready for your project! 1. Introduction to NetworkX. Introduction to NetworkX - network analysis Vast amounts of network data are being generated and collected • Sociology: web pages, mobile phones, social networks • Technology: Internet routers, vehicular flows, power grids How can we analyze this networks? Introduction to NetworkX - Python awesomeness Introduction to NetworkX “Python package for the creation, manipulation and study of the structure, dynamics and functions of complex networks.” • Data structures for representing many types of networks, or graphs • Nodes can be any (hashable) Python object, edges can contain arbitrary data • Flexibility ideal for representing networks found in many different fields • Easy to install on multiple platforms • Online up-to-date documentation • First public release in April 2005 Introduction to NetworkX - design requirements • Tool to study the structure and dynamics of social, biological, and infrastructure networks • Ease-of-use and rapid development in a collaborative, multidisciplinary environment • Easy to learn, easy to teach • Open-source tool base that can easily grow in a multidisciplinary environment with non-expert users -
Counting Independent Sets in Graphs with Bounded Bipartite Pathwidth∗
Counting independent sets in graphs with bounded bipartite pathwidth∗ Martin Dyery Catherine Greenhillz School of Computing School of Mathematics and Statistics University of Leeds UNSW Sydney, NSW 2052 Leeds LS2 9JT, UK Australia [email protected] [email protected] Haiko M¨uller∗ School of Computing University of Leeds Leeds LS2 9JT, UK [email protected] 7 August 2019 Abstract We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalisation of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights. 1 Introduction There is a well-known bijection between matchings of a graph G and independent sets in the line graph of G. We will show that we can approximate the number of independent sets ∗A preliminary version of this paper appeared as [19]. yResearch supported by EPSRC grant EP/S016562/1 \Sampling in hereditary classes". zResearch supported by Australian Research Council grant DP190100977. 1 in graphs for which all bipartite induced subgraphs are well structured, in a sense that we will define precisely. -
1 Hamiltonian Path
6.S078 Fine-Grained Algorithms and Complexity MIT Lecture 17: Algorithms for Finding Long Paths (Part 1) November 2, 2020 In this lecture and the next, we will introduce a number of algorithmic techniques used in exponential-time and FPT algorithms, through the lens of one parametric problem: Definition 0.1 (k-Path) Given a directed graph G = (V; E) and parameter k, is there a simple path1 in G of length ≥ k? Already for this simple-to-state problem, there are quite a few radically different approaches to solving it faster; we will show you some of them. We’ll see algorithms for the case of k = n (Hamiltonian Path) and then we’ll turn to “parameterizing” these algorithms so they work for all k. A number of papers in bioinformatics have used quick algorithms for k-Path and related problems to analyze various networks that arise in biology (some references are [SIKS05, ADH+08, YLRS+09]). In the following, we always denote the number of vertices jV j in our given graph G = (V; E) by n, and the number of edges jEj by m. We often associate the set of vertices V with the set [n] := f1; : : : ; ng. 1 Hamiltonian Path Before discussing k-Path, it will be useful to first discuss algorithms for the famous NP-complete Hamiltonian path problem, which is the special case where k = n. Essentially all algorithms we discuss here can be adapted to obtain algorithms for k-Path! The naive algorithm for Hamiltonian Path takes time about n! = 2Θ(n log n) to try all possible permutations of the nodes (which can also be adapted to get an O?(k!)-time algorithm for k-Path, as we’ll see). -
General Approach to Line Graphs of Graphs 1
DEMONSTRATIO MATHEMATICA Vol. XVII! No 2 1985 Antoni Marczyk, Zdzislaw Skupien GENERAL APPROACH TO LINE GRAPHS OF GRAPHS 1. Introduction A unified approach to the notion of a line graph of general graphs is adopted and proofs of theorems announced in [6] are presented. Those theorems characterize five different types of line graphs. Both Krausz-type and forbidden induced sub- graph characterizations are provided. So far other authors introduced and dealt with single spe- cial notions of a line graph of graphs possibly belonging to a special subclass of graphs. In particular, the notion of a simple line graph of a simple graph is implied by a paper of Whitney (1932). Since then it has been repeatedly introduc- ed, rediscovered and generalized by many authors, among them are Krausz (1943), Izbicki (1960$ a special line graph of a general graph), Sabidussi (1961) a simple line graph of a loop-free graph), Menon (1967} adjoint graph of a general graph) and Schwartz (1969; interchange graph which coincides with our line graph defined below). In this paper we follow another way, originated in our previous work [6]. Namely, we distinguish special subclasses of general graphs and consider five different types of line graphs each of which is defined in a natural way. Note that a similar approach to the notion of a line graph of hypergraphs can be adopted. We consider here the following line graphsi line graphs, loop-free line graphs, simple line graphs, as well as augmented line graphs and augmented loop-free line graphs. - 447 - 2 A. Marczyk, Z. -
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. -
Efficient Domination and Polarity
Efficient Domination and Polarity Dissertation zur Erlangung des akademischen Grades Doktor-Ingenieur (Dr.-Ing.) Universität Rostock Fakultät für Informatik und Elektrotechnik vorgelegt von Ragnar Christopher Nevries Gutachter: Prof. Dr. Andreas Brandstädt, Institut für Informatik, Universität Rostock Prof. Dr. Peter Widmayer, Institut für Theoretische Informatik, ETH Zürich Prof. Dr. Jing Huang, Mathematics and Statistics, University of Victoria eingereicht am 5. Februar 2014 verteidigt am 25. Juli 2014 Abstract This thesis considers Efficient Domination, Efficient Edge Domination, Polar- ity, and Monopolarity, graph problems that ask for a vertex or edge subset that is a packing and a covering at the same time. Efficient Domination seeks for an independent vertex subset D such that all other vertices have exactly one neighbor in D. Here, packing means that the vertices of D must not be too close to each other and, in contrast, covering requires that they have to be near to the other vertices. Efficient Edge Domination is the edge version of Efficient Domination. Polarity asks for a vertex subset that induces a complete multipartite graph—the packing aspect—and that contains a vertex of every induced P3—the covering aspect. Monopolarity is the special case of Polarity where the complete multipartite graph has to be edgeless. Since all these problems are NP-complete in general, for each problem a lot of effort has been put into separating the graph classes on which the problem remains NP-complete from those that admit an efficient algorithm. This thesis pursues both directions. On the one hand, we introduce a framework for our NP-completeness proofs and use it to sharpen known results for all mentioned problems. -
K-Path Centrality: a New Centrality Measure in Social Networks
Centrality Metrics in Social Network Analysis K-path: A New Centrality Metric Experiments Summary K-Path Centrality: A New Centrality Measure in Social Networks Adriana Iamnitchi University of South Florida joint work with Tharaka Alahakoon, Rahul Tripathi, Nicolas Kourtellis and Ramanuja Simha Adriana Iamnitchi K-Path Centrality: A New Centrality Measure in Social Networks 1 of 23 Centrality Metrics in Social Network Analysis Centrality Metrics Overview K-path: A New Centrality Metric Betweenness Centrality Experiments Applications Summary Computing Betweenness Centrality Centrality Metrics in Social Network Analysis Betweenness Centrality - how much a node controls the flow between any other two nodes Closeness Centrality - the extent a node is near all other nodes Degree Centrality - the number of ties to other nodes Eigenvector Centrality - the relative importance of a node Adriana Iamnitchi K-Path Centrality: A New Centrality Measure in Social Networks 2 of 23 Centrality Metrics in Social Network Analysis Centrality Metrics Overview K-path: A New Centrality Metric Betweenness Centrality Experiments Applications Summary Computing Betweenness Centrality Betweenness Centrality measures the extent to which a node lies on the shortest path between two other nodes betweennes CB (v) of a vertex v is the summation over all pairs of nodes of the fractional shortest paths going through v. Definition (Betweenness Centrality) For every vertex v 2 V of a weighted graph G(V ; E), the betweenness centrality CB (v) of v is defined by X X σst (v) CB -
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. -
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.