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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. -
Graph Varieties Axiomatized by Semimedial, Medial, and Some Other Groupoid Identities
Discussiones Mathematicae General Algebra and Applications 40 (2020) 143–157 doi:10.7151/dmgaa.1344 GRAPH VARIETIES AXIOMATIZED BY SEMIMEDIAL, MEDIAL, AND SOME OTHER GROUPOID IDENTITIES Erkko Lehtonen Technische Universit¨at Dresden Institut f¨ur Algebra 01062 Dresden, Germany e-mail: [email protected] and Chaowat Manyuen Department of Mathematics, Faculty of Science Khon Kaen University Khon Kaen 40002, Thailand e-mail: [email protected] Abstract Directed graphs without multiple edges can be represented as algebras of type (2, 0), so-called graph algebras. A graph is said to satisfy an identity if the corresponding graph algebra does, and the set of all graphs satisfying a set of identities is called a graph variety. We describe the graph varieties axiomatized by certain groupoid identities (medial, semimedial, autodis- tributive, commutative, idempotent, unipotent, zeropotent, alternative). Keywords: graph algebra, groupoid, identities, semimediality, mediality. 2010 Mathematics Subject Classification: 05C25, 03C05. 1. Introduction Graph algebras were introduced by Shallon [10] in 1979 with the purpose of providing examples of nonfinitely based finite algebras. Let us briefly recall this concept. Given a directed graph G = (V, E) without multiple edges, the graph algebra associated with G is the algebra A(G) = (V ∪ {∞}, ◦, ∞) of type (2, 0), 144 E. Lehtonen and C. Manyuen where ∞ is an element not belonging to V and the binary operation ◦ is defined by the rule u, if (u, v) ∈ E, u ◦ v := (∞, otherwise, for all u, v ∈ V ∪ {∞}. We will denote the product u ◦ v simply by juxtaposition uv. Using this representation, we may view any algebraic property of a graph algebra as a property of the graph with which it is associated. -
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). -
Graph Theory 1 Introduction
6.042/18.062J Mathematics for Computer Science March 1, 2005 Srini Devadas and Eric Lehman Lecture Notes Graph Theory 1 Introduction Informally, a graph is a bunch of dots connected by lines. Here is an example of a graph: B H D A F G I C E Sadly, this definition is not precise enough for mathematical discussion. Formally, a graph is a pair of sets (V, E), where: • Vis a nonempty set whose elements are called vertices. • Eis a collection of twoelement subsets of Vcalled edges. The vertices correspond to the dots in the picture, and the edges correspond to the lines. Thus, the dotsandlines diagram above is a pictorial representation of the graph (V, E) where: V={A, B, C, D, E, F, G, H, I} E={{A, B} , {A, C} , {B, D} , {C, D} , {C, E} , {E, F } , {E, G} , {H, I}} . 1.1 Definitions A nuisance in first learning graph theory is that there are so many definitions. They all correspond to intuitive ideas, but can take a while to absorb. Some ideas have multi ple names. For example, graphs are sometimes called networks, vertices are sometimes called nodes, and edges are sometimes called arcs. Even worse, no one can agree on the exact meanings of terms. For example, in our definition, every graph must have at least one vertex. However, other authors permit graphs with no vertices. (The graph with 2 Graph Theory no vertices is the single, stupid counterexample to many wouldbe theorems— so we’re banning it!) This is typical; everyone agrees moreorless what each term means, but dis agrees about weird special cases. -
Arxiv:2006.06067V2 [Math.CO] 4 Jul 2021
Treewidth versus clique number. I. Graph classes with a forbidden structure∗† Cl´ement Dallard1 Martin Milaniˇc1 Kenny Storgelˇ 2 1 FAMNIT and IAM, University of Primorska, Koper, Slovenia 2 Faculty of Information Studies, Novo mesto, Slovenia [email protected] [email protected] [email protected] Treewidth is an important graph invariant, relevant for both structural and algo- rithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call (tw,ω)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, in- duced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is (tw,ω)-bounded. Our results yield an infinite family of χ-bounded induced-minor-closed graph classes and imply that the class of 1-perfectly orientable graphs is (tw,ω)-bounded, leading to linear-time algorithms for k-coloring 1-perfectly orientable graphs for every fixed k. This answers a question of Breˇsar, Hartinger, Kos, and Milaniˇc from 2018, and one of Beisegel, Chudnovsky, Gurvich, Milaniˇc, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of (tw,ω)- boundedness related to list k-coloring and clique problems. In addition, we propose a question about the complexity of the Maximum Weight Independent Set prob- lem in (tw,ω)-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor. -
HABILITATION THESIS Petr Gregor Combinatorial Structures In
HABILITATION THESIS Petr Gregor Combinatorial Structures in Hypercubes Computer Science - Theoretical Computer Science Prague, Czech Republic April 2019 Contents Synopsis of the thesis4 List of publications in the thesis7 1 Introduction9 1.1 Hypercubes...................................9 1.2 Queue layouts.................................. 14 1.3 Level-disjoint partitions............................ 19 1.4 Incidence colorings............................... 24 1.5 Distance magic labelings............................ 27 1.6 Parity vertex colorings............................. 29 1.7 Gray codes................................... 30 1.8 Linear extension diameter........................... 36 Summary....................................... 38 2 Queue layouts 51 3 Level-disjoint partitions 63 4 Incidence colorings 125 5 Distance magic labelings 143 6 Parity vertex colorings 149 7 Gray codes 155 8 Linear extension diameter 235 3 Synopsis of the thesis The thesis is compiled as a collection of 12 selected publications on various combinatorial structures in hypercubes accompanied with a commentary in the introduction. In these publications from years between 2012 and 2018 we solve, in some cases at least partially, several open problems or we significantly improve previously known results. The list of publications follows after the synopsis. The thesis is organized into 8 chapters. Chapter 1 is an umbrella introduction that contains background, motivation, and summary of the most interesting results. Chapter 2 studies queue layouts of hypercubes. A queue layout is a linear ordering of vertices together with a partition of edges into sets, called queues, such that in each set no two edges are nested with respect to the ordering. The results in this chapter significantly improve previously known upper and lower bounds on the queue-number of hypercubes associated with these layouts. -
HAMILTONICITY in CAYLEY GRAPHS and DIGRAPHS of FINITE ABELIAN GROUPS. Contents 1. Introduction. 1 2. Graph Theory: Introductory
HAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS. MARY STELOW Abstract. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown. Several results are then pre- sented: firstly, it is shown that if G is an abelian group, then G has a Cayley digraph with a Hamiltonian cycle. It is then proven that every Cayley di- graph of a Dedekind group has a Hamiltonian path. Finally, we show that all Cayley graphs of Abelian groups have Hamiltonian cycles. Further results, applications, and significance of the study of Hamiltonicity of Cayley graphs and digraphs are then discussed. Contents 1. Introduction. 1 2. Graph Theory: Introductory Definitions. 2 3. Cayley Graphs and Digraphs. 2 4. Hamiltonian Cycles in Cayley Digraphs of Finite Abelian Groups 5 5. Hamiltonian Paths in Cayley Digraphs of Dedekind Groups. 7 6. Cayley Graphs of Finite Abelian Groups are Guaranteed a Hamiltonian Cycle. 8 7. Conclusion; Further Applications and Research. 9 8. Acknowledgements. 9 References 10 1. Introduction. The topic of Cayley digraphs and graphs exhibits an interesting and important intersection between the world of groups, group theory, and abstract algebra and the world of graph theory and combinatorics. In this paper, I aim to highlight this intersection and to introduce an area in the field for which many basic problems re- main open.The theorems presented are taken from various discrete math journals. Here, these theorems are analyzed and given lengthier treatment in order to be more accessible to those without much background in group or graph theory. -
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. -
Treewidth-Erickson.Pdf
Computational Topology (Jeff Erickson) Treewidth Or il y avait des graines terribles sur la planète du petit prince . c’étaient les graines de baobabs. Le sol de la planète en était infesté. Or un baobab, si l’on s’y prend trop tard, on ne peut jamais plus s’en débarrasser. Il encombre toute la planète. Il la perfore de ses racines. Et si la planète est trop petite, et si les baobabs sont trop nombreux, ils la font éclater. [Now there were some terrible seeds on the planet that was the home of the little prince; and these were the seeds of the baobab. The soil of that planet was infested with them. A baobab is something you will never, never be able to get rid of if you attend to it too late. It spreads over the entire planet. It bores clear through it with its roots. And if the planet is too small, and the baobabs are too many, they split it in pieces.] — Antoine de Saint-Exupéry (translated by Katherine Woods) Le Petit Prince [The Little Prince] (1943) 11 Treewidth In this lecture, I will introduce a graph parameter called treewidth, that generalizes the property of having small separators. Intuitively, a graph has small treewidth if it can be recursively decomposed into small subgraphs that have small overlap, or even more intuitively, if the graph resembles a ‘fat tree’. Many problems that are NP-hard for general graphs can be solved in polynomial time for graphs with small treewidth. Graphs embedded on surfaces of small genus do not necessarily have small treewidth, but they can be covered by a small number of subgraphs, each with small treewidth. -
The Cube of Every Connected Graph Is 1-Hamiltonian
JOURNAL OF RESEARCH of the National Bureau of Standards- B. Mathematical Sciences Vol. 73B, No.1, January- March 1969 The Cube of Every Connected Graph is l-Hamiltonian * Gary Chartrand and S. F. Kapoor** (November 15,1968) Let G be any connected graph on 4 or more points. The graph G3 has as its point set that of C, and two distinct pointE U and v are adjacent in G3 if and only if the distance betwee n u and v in G is at most three. It is shown that not only is G" hamiltonian, but the removal of any point from G" still yields a hamiltonian graph. Key Words: C ube of a graph; graph; hamiltonian. Let G be a graph (finite, undirected, with no loops or multiple lines). A waLk of G is a finite alternating sequence of points and lines of G, beginning and ending with a point and where each line is incident with the points immediately preceding and followin g it. A walk in which no point is repeated is called a path; the Length of a path is the number of lines in it. A graph G is connected if between every pair of distinct points there exists a path, and for such a graph, the distance between two points u and v is defined as the length of the shortest path if u 0;1= v and zero if u = v. A walk with at least three points in which the first and last points are the same but all other points are dis tinct is called a cycle. -
A Survey of Line Graphs and Hamiltonian Paths Meagan Field
University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-7-2012 A Survey of Line Graphs and Hamiltonian Paths Meagan Field Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Field, Meagan, "A Survey of Line Graphs and Hamiltonian Paths" (2012). Math Theses. Paper 3. http://hdl.handle.net/10950/86 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. A SURVEY OF LINE GRAPHS AND HAMILTONIAN PATHS by MEAGAN FIELD A thesis submitted in partial fulfillment of the requirements for the degree of Master’s of Science Department of Mathematics Christina Graves, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2012 Contents List of Figures .................................. ii Abstract ...................................... iii 1 Definitions ................................... 1 2 Background for Hamiltonian Paths and Line Graphs ........ 11 3 Proof of Theorem 2.2 ............................ 17 4 Conclusion ................................... 34 References ..................................... 35 i List of Figures 1GraphG..................................1 2GandL(G)................................3 3Clawgraph................................4 4Claw-freegraph..............................4 5Graphwithaclawasaninducedsubgraph...............4 6GraphP .................................. 7 7AgraphCcontainingcomponents....................8 8Asimplegraph,Q............................9 9 First and second step using R2..................... 9 10 Third and fourth step using R2..................... 10 11 Graph D and its reduced graph R(D).................. 10 12 S5 and its line graph, K5 ......................... 13 13 S8 and its line graph, K8 ......................... 14 14 Line graph of graph P ......................... -
Subgraph Isomorphism in Graph Classes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 312 (2012) 3164–3173 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Subgraph isomorphism in graph classes Shuji Kijima a, Yota Otachi b, Toshiki Saitoh c,∗, Takeaki Uno d a Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka, 819-0395, Japan b School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa, 923-1292, Japan c Graduate School of Engineering, Kobe University, Kobe, 657-8501, Japan d National Institute of Informatics, Tokyo, 101-8430, Japan article info a b s t r a c t Article history: We investigate the computational complexity of the following restricted variant of Received 17 September 2011 Subgraph Isomorphism: given a pair of connected graphs G D .VG; EG/ and H D .VH ; EH /, Received in revised form 6 July 2012 determine if H is isomorphic to a spanning subgraph of G. The problem is NP-complete in Accepted 9 July 2012 general, and thus we consider cases where G and H belong to the same graph class such as Available online 28 July 2012 the class of proper interval graphs, of trivially perfect graphs, and of bipartite permutation graphs. For these graph classes, several restricted versions of Subgraph Isomorphism Keywords: such as , , , and can be solved Subgraph isomorphism Hamiltonian Path Clique Bandwidth Graph Isomorphism Graph class in polynomial time, while these problems are hard in general. NP-completeness ' 2012 Elsevier B.V.