THE CHROMATIC POLYNOMIAL 1. Introduction a Common Problem in the Study of Graph Theory Is Coloring the Vertices of a Graph So Th
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On Treewidth and Graph Minors
On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx. -
CHROMATIC POLYNOMIALS of PLANE TRIANGULATIONS 1. Basic
1990, 2005 D. R. Woodall, School of Mathematical Sciences, University of Nottingham CHROMATIC POLYNOMIALS OF PLANE TRIANGULATIONS 1. Basic results. Throughout this survey, G will denote a multigraph with n vertices, m edges, c components and b blocks, and m′ will denote the smallest number of edges whose deletion from G leaves a simple graph. The corresponding numbers for Gi will be ′ denoted by ni , mi , ci , bi and mi . Let P(G, t) denote the number of different (proper vertex-) t-colourings of G. Anticipating a later result, we call P(G, t) the chromatic polynomial of G. It was introduced by G. D. Birkhoff (1912), who proved many of the following basic results. Proposition 1. (Examples.) Here Tn denotes an arbitrary tree with n vertices, Fn denotes an arbitrary forest with n vertices and c components, and Rn denotes the graph of an arbitrary triangulated polygon with n vertices: that is, a plane n-gon divided into triangles by n − 3 noncrossing chords. = n (a) P(Kn , t) t , = − n −1 (b) P(Tn , t) t(t 1) , = − − n −2 (c) P(Rn , t) t(t 1)(t 2) , = − − − + (d) P(Kn , t) t(t 1)(t 2)...(t n 1), = c − n −c (e) P(Fn , t) t (t 1) , = − n + − n − (f) P(Cn , t) (t 1) ( 1) (t 1) − = (−1)nt(t − 1)[1 + (1 − t) + (1 − t)2 + ... + (1 − t)n 2]. Proposition 2. (a) If the components of G are G1,..., Gc , then = P(G, t) P(G1, t)...P(Gc , t). = ∪ ∩ = (b) If G G1 G2 where G1 G2 Kr , then P(G1, t) P(G2, t) ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡£¡¢¡¢¡¢¡ P(G, t) = ¡ . -
Approximating the Chromatic Polynomial of a Graph
Approximating the Chromatic Polynomial Yvonne Kemper1 and Isabel Beichl1 Abstract Chromatic polynomials are important objects in graph theory and statistical physics, but as a result of computational difficulties, their study is limited to graphs that are small, highly structured, or very sparse. We have devised and implemented two algorithms that approximate the coefficients of the chromatic polynomial P (G, x), where P (G, k) is the number of proper k-colorings of a graph G for k ∈ N. Our algorithm is based on a method of Knuth that estimates the order of a search tree. We compare our results to the true chro- matic polynomial in several known cases, and compare our error with previous approximation algorithms. 1 Introduction The chromatic polynomial P (G, x) of a graph G has received much atten- tion as a result of the now-resolved four-color problem, but its relevance extends beyond combinatorics, and its domain beyond the natural num- bers. To start, the chromatic polynomial is related to the Tutte polynomial, and evaluating these polynomials at different points provides information about graph invariants and structure. In addition, P (G, x) is central in applications such as scheduling problems [26] and the q-state Potts model in statistical physics [10, 25]. The former occur in a variety of contexts, from algorithm design to factory procedures. For the latter, the relation- ship between P (G, x) and the Potts model connects statistical mechanics and graph theory, allowing researchers to study phenomena such as the behavior of ferromagnets. Unfortunately, computing P (G, x) for a general graph G is known to be #P -hard [16, 22] and deciding whether or not a graph is k-colorable is NP -hard [11]. -
Exclusive Graph Searching Lélia Blin, Janna Burman, Nicolas Nisse
Exclusive Graph Searching Lélia Blin, Janna Burman, Nicolas Nisse To cite this version: Lélia Blin, Janna Burman, Nicolas Nisse. Exclusive Graph Searching. Algorithmica, Springer Verlag, 2017, 77 (3), pp.942-969. 10.1007/s00453-016-0124-0. hal-01266492 HAL Id: hal-01266492 https://hal.archives-ouvertes.fr/hal-01266492 Submitted on 2 Feb 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. Exclusive Graph Searching∗ L´eliaBlin Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, Universit´ed'Evry-Val-d'Essonne. LIP6 UMR 7606, 4 place Jussieu 75005, Paris, France [email protected] Janna Burman LRI, Universit´eParis Sud, CNRS, UMR-8623, France. [email protected] Nicolas Nisse Inria, France. Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France. [email protected] February 2, 2016 Abstract This paper tackles the well known graph searching problem, where a team of searchers aims at capturing an intruder in a network, modeled as a graph. This problem has been mainly studied for its relationship with the pathwidth of graphs. All variants of this problem assume that any node can be simultaneously occupied by several searchers. -
Computing Tutte Polynomials
Computing Tutte Polynomials Gary Haggard1, David J. Pearce2, and Gordon Royle3 1 Bucknell University [email protected] 2 Computer Science Group, Victoria University of Wellington, [email protected] 3 School of Mathematics and Statistics, University of Western Australia [email protected] Abstract. The Tutte polynomial of a graph, also known as the partition function of the q-state Potts model, is a 2-variable polynomial graph in- variant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chro- matic polynomial and flow polynomial as partial evaluations, and various numerical invariants such as the number of spanning trees as complete evaluations. However despite its ubiquity, there are no widely-available effective computational tools able to compute the Tutte polynomial of a general graph of reasonable size. In this paper we describe the implemen- tation of a program that exploits isomorphisms in the computation tree to extend the range of graphs for which it is feasible to compute their Tutte polynomials. We also consider edge-selection heuristics which give good performance in practice. We empirically demonstrate the utility of our program on random graphs. More evidence of its usefulness arises from our success in finding counterexamples to a conjecture of Welsh on the location of the real flow roots of a graph. 1 Introduction The Tutte polynomial of a graph is a 2-variable polynomial of significant im- portance in mathematics, statistical physics and biology [25]. In a strong sense it “contains” every graphical invariant that can be computed by deletion and contraction. -
Blast Domination for Mycielski's Graph of Graphs
International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-8, Issue-6S3, September 2019 Blast Domination for Mycielski’s Graph of Graphs K. Ameenal Bibi, P.Rajakumari AbstractThe hub of this article is a search on the behavior of is the minimum cardinality of a distance-2 dominating set in the Blast domination and Blast distance-2 domination for 퐺. Mycielski’s graph of some particular graphs and zero divisor graphs. Definition 2.3[7] A non-empty subset 퐷 of 푉 of a connected graph 퐺 is Key Words:Blast domination number, Blast distance-2 called a Blast dominating set, if 퐷 is aconnected dominating domination number, Mycielski’sgraph. set and the induced sub graph < 푉 − 퐷 >is triple connected. The minimum cardinality taken over all such Blast I. INTRODUCTION dominating sets is called the Blast domination number of The concept of triple connected graphs was introduced by 퐺and is denoted by tc . Paulraj Joseph et.al [9]. A graph is said to be triple c (G) connected if any three vertices lie on a path in G. In [6] the Definition2.4 authors introduced triple connected domination number of a graph. A subset D of V of a nontrivial graph G is said to A non-empty subset 퐷 of vertices in a graph 퐺 is a blast betriple connected dominating set, if D is a dominating set distance-2 dominating set if every vertex in 푉 − 퐷 is within and <D> is triple connected. The minimum cardinality taken distance-2 of atleast one vertex in 퐷. -
Empirical Comparison of Greedy Strategies for Learning Markov Networks of Treewidth K
Empirical Comparison of Greedy Strategies for Learning Markov Networks of Treewidth k K. Nunez, J. Chen and P. Chen G. Ding and R. F. Lax B. Marx Dept. of Computer Science Dept. of Mathematics Dept. of Experimental Statistics LSU LSU LSU Baton Rouge, LA 70803 Baton Rouge, LA 70803 Baton Rouge, LA 70803 Abstract—We recently proposed the Edgewise Greedy Algo- simply one less than the maximum of the orders of the rithm (EGA) for learning a decomposable Markov network of maximal cliques [5], a treewidth k DMN approximation to treewidth k approximating a given joint probability distribution a joint probability distribution of n random variables offers of n discrete random variables. The main ingredient of our algorithm is the stepwise forward selection algorithm (FSA) due computational benefits as the approximation uses products of to Deshpande, Garofalakis, and Jordan. EGA is an efficient alter- marginal distributions each involving at most k + 1 of the native to the algorithm (HGA) by Malvestuto, which constructs random variables. a model of treewidth k by selecting hyperedges of order k +1. In The treewidth k DNM learning problem deals with a given this paper, we present results of empirical studies that compare empirical distribution P ∗ (which could be given in the form HGA, EGA and FSA-K which is a straightforward application of FSA, in terms of approximation accuracy (measured by of a distribution table or in the form of a set of training data), KL-divergence) and computational time. Our experiments show and the objective is to find a DNM (i.e., a chordal graph) that (1) on the average, all three algorithms produce similar G of treewidth k which defines a probability distribution Pµ approximation accuracy; (2) EGA produces comparable or better that ”best” approximates P ∗ according to some criterion. -
Determination and Testing the Domination Numbers of Tadpole Graph, Book Graph and Stacked Book Graph Using MATLAB Ayhan A
College of Basic Education Researchers Journal Vol. 10, No. 1 Determination and Testing the Domination Numbers of Tadpole Graph, Book Graph and Stacked Book Graph Using MATLAB Ayhan A. khalil Omar A.Khalil Technical College of Mosul -Foundation of Technical Education Received: 11/4/2010 ; Accepted: 24/6/2010 Abstract: A set is said to be dominating set of G if for every there exists a vertex such that . The minimum cardinality of vertices among dominating set of G is called the domination number of G denoted by . We investigate the domination number of Tadpole graph, Book graph and Stacked Book graph. Also we test our theoretical results in computer by introducing a matlab procedure to find the domination number , dominating set S and draw this graph that illustrates the vertices of domination this graphs. It is proved that: . Keywords: Dominating set, Domination number, Tadpole graph, Book graph, stacked graph. إﯾﺟﺎد واﺧﺗﺑﺎر اﻟﻌدد اﻟﻣﮭﯾﻣن(اﻟﻣﺳﯾطر)Domination number ﻟﻠﺑﯾﺎﻧﺎت ﺗﺎدﺑول (Tadpole graph)، ﺑﯾﺎن ﺑوك (Book graph) وﺑﯾﺎن ﺳﺗﺎﻛﯾت ﺑوك (Stacked Book graph) ﺑﺎﺳﺗﻌﻣﺎل اﻟﻣﺎﺗﻼب أﯾﮭﺎن اﺣﻣد ﺧﻠﯾل ﻋﻣر اﺣﻣد ﺧﻠﯾل اﻟﻛﻠﯾﺔ اﻟﺗﻘﻧﯾﺔ-ﻣوﺻل- ھﯾﺋﺔ اﻟﺗﻌﻠﯾم اﻟﺗﻘﻧﻲ ﻣﻠﺧص اﻟﺑﺣث : ﯾﻘﺎل ﻷﯾﺔ ﻣﺟﻣوﻋﺔ ﺟزﺋﯾﺔ S ﻣن ﻣﺟﻣوﻋﺔ اﻟرؤوس V ﻓﻲ ﺑﯾﺎن G ﺑﺄﻧﻬﺎ ﻣﺟﻣوﻋﺔ ﻣﻬﯾﻣﻧﺔ Dominating set إذا ﻛﺎن ﻟﻛل أرس v ﻓﻲ اﻟﻣﺟﻣوﻋﺔ V-S ﯾوﺟد أرس u ﻓﻲS ﺑﺣﯾث uv ﺿﻣن 491 Ayhan A. & Omar A. ﺣﺎﻓﺎت اﻟﺑﯾﺎن،. وﯾﻌرف اﻟﻌدد اﻟﻣﻬﯾﻣن Domination number ﺑﺄﻧﻪ أﺻﻐر ﻣﺟﻣوﻋﺔ أﺳﺎﺳﯾﺔ ﻣﻬﯾﻣﻧﺔ Domination. ﻓﻲ ﻫذا اﻟﺑﺣث ﺳوف ﻧدرس اﻟﻌدد اﻟﻣﻬﯾﻣن Domination number ﻟﺑﯾﺎن ﺗﺎدﺑول (Tadpole graph) ، ﺑﯾﺎن ﺑوك (Book graph) وﺑﯾﺎن ﺳﺗﺎﻛﯾت ﺑوك ( Stacked Book graph). -
Discrete Mathematics Rooted Directed Path Graphs Are Leaf Powers
Discrete Mathematics 310 (2010) 897–910 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Rooted directed path graphs are leaf powers Andreas Brandstädt a, Christian Hundt a, Federico Mancini b, Peter Wagner a a Institut für Informatik, Universität Rostock, D-18051, Germany b Department of Informatics, University of Bergen, N-5020 Bergen, Norway article info a b s t r a c t Article history: Leaf powers are a graph class which has been introduced to model the problem of Received 11 March 2009 reconstructing phylogenetic trees. A graph G D .V ; E/ is called k-leaf power if it admits a Received in revised form 2 September 2009 k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance Accepted 13 October 2009 between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a Available online 30 October 2009 k-leaf power for some k 2 N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a Keywords: given graph is a k-leaf power. Leaf powers Leaf roots We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a Graph class inclusions well-known graph class with absolutely different motivations. After this, we provide the Strongly chordal graphs largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path Fixed tolerance NeST graphs graphs. -
The Chromatic Polynomial
Eotv¨ os¨ Lorand´ University Faculty of Science Master's Thesis The chromatic polynomial Author: Supervisor: Tam´asHubai L´aszl´oLov´aszPhD. MSc student in mathematics professor, Comp. Sci. Department [email protected] [email protected] 2009 Abstract After introducing the concept of the chromatic polynomial of a graph, we describe its basic properties and present a few examples. We continue with observing how the co- efficients and roots relate to the structure of the underlying graph, with emphasis on a theorem by Sokal bounding the complex roots based on the maximal degree. We also prove an improved version of this theorem. Finally we look at the Tutte polynomial, a generalization of the chromatic polynomial, and some of its applications. Acknowledgements I am very grateful to my supervisor, L´aszl´oLov´asz,for this interesting subject and his assistance whenever I needed. I would also like to say thanks to M´artonHorv´ath, who pointed out a number of mistakes in the first version of this thesis. Contents 0 Introduction 4 1 Preliminaries 5 1.1 Graph coloring . .5 1.2 The chromatic polynomial . .7 1.3 Deletion-contraction property . .7 1.4 Examples . .9 1.5 Constructions . 10 2 Algebraic properties 13 2.1 Coefficients . 13 2.2 Roots . 15 2.3 Substitutions . 18 3 Bounding complex roots 19 3.1 Motivation . 19 3.2 Preliminaries . 19 3.3 Proving the theorem . 20 4 Tutte's polynomial 23 4.1 Flow polynomial . 23 4.2 Reliability polynomial . 24 4.3 Statistical mechanics and the Potts model . 24 4.4 Jones polynomial of alternating knots . -
Constructing Arbitrarily Large Graphs with a Specified Number Of
Electronic Journal of Graph Theory and Applications 4 (1) (2019), 1–8 Constructing Arbitrarily Large Graphs with a Specified Number of Hamiltonian Cycles Michael Haythorpea aSchool of Computer Science, Engineering and Mathematics, Flinders University, 1284 South Road, Clovelly Park, SA 5042, Australia michael.haythorpe@flinders.edu.au Abstract A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing 5n − 9 vertices and k Hamiltonian cycles for any choice of integers n ≥ k ≥ 4. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when k is chosen to be much smaller than n. Keywords: Hamiltonian cycles, Graph Construction, Broken Crown Mathematics Subject Classification : 05C45 1. Introduction The Hamiltonian cycle problem (HCP) is a famous NP-complete problem in which one must determine whether a given graph contains a simple cycle traversing all vertices of the graph, or not. Such a simple cycle is called a Hamiltonian cycle (HC), and a graph containing at least one arXiv:1902.10351v1 [math.CO] 27 Feb 2019 Hamiltonian cycle is said to be a Hamiltonian graph. Typically, randomly generated graphs (such as Erdos-R˝ enyi´ graphs), if connected, are Hamil- tonian and contain many Hamiltonian cycles. Although HCP is an NP-complete problem, for these graphs it is often fairly easy for a sophisticated heuristic (e.g. see Concorde [1], Keld Helsgaun’s LKH [4] or Snakes-and-ladders Heuristic [2]) to discover one of the multitude of Hamiltonian cy- cles through a clever search. -
Dominating Cycles in Halin Graphs*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 86 (1990) 215-224 215 North-Holland DOMINATING CYCLES IN HALIN GRAPHS* Mirosiawa SKOWRONSKA Institute of Mathematics, Copernicus University, Chopina 12/18, 87-100 Torun’, Poland Maciej M. SYStO Institute of Computer Science, University of Wroclaw, Przesmyckiego 20, 51-151 Wroclaw, Poland Received 2 December 1988 A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D-cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Hahn graph; minor modifications can give formulae for finding a shortest D-cycle. Then, dominating cycles and D-cycles in a Halin graph H are characterized in terms of the cycle graph, the intersection graph of the faces of H. 1. Preliminaries The various domination problems have been extensively studied. Among them is the problem whether a graph has a dominating cycle. All graphs in this paper have no loops and multiple edges. A dominating cycle in a graph G = (V(G), E(G)) is a subgraph C of G which is a cycle and every vertex of V(G) \ V(C) is adjacent to a vertex of C. There are graphs which have no dominating cycles, and moreover, determining whether a graph has a dominating cycle on at most 1 vertices is NP-complete even in the class of planar graphs [7], chordal, bipartite and split graphs [3].