A Charming Class of Perfectly Orderable Graphs
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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. -
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. -
Computing Directed Path-Width and Directed Tree-Width of Recursively
Computing directed path-width and directed tree-width of recursively defined digraphs∗ Frank Gurski1 and Carolin Rehs1 1University of D¨usseldorf, Institute of Computer Science, Algorithmics for Hard Problems Group, 40225 D¨usseldorf, Germany June 13, 2018 Abstract In this paper we consider the directed path-width and directed tree-width of recursively defined digraphs. As an important combinatorial tool, we show how the directed path-width and the directed tree-width can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs. These results imply the equality of directed path-width and directed tree-width for all digraphs which can be defined by these four operations. This allows us to show a linear-time solution for computing the directed path-width and directed tree-width of all these digraphs. Since directed co-graphs are precisely those digraphs which can be defined by the disjoint union, order composition, and series composition our results imply the equality of directed path-width and directed tree-width for directed co-graphs and also a linear-time solution for computing the directed path-width and directed tree-width of directed co-graphs, which generalizes the known results for undirected co-graphs of Bodlaender and M¨ohring. Keywords: directed path-width; directed tree-width; directed co-graphs 1 Introduction Tree-width is a well-known graph parameter [36]. Many NP-hard graph problems admit poly- nomial-time solutions when restricted to graphs of bounded tree-width using the tree-decom- position [1, 3, 22, 27]. The same holds for path-width [35] since a path-decomposition can be regarded as a special case of a tree-decomposition. -
Online Graph Coloring
Online Graph Coloring Jinman Zhao - CSC2421 Online Graph coloring Input sequence: Output: Goal: Minimize k. k is the number of color used. Chromatic number: Smallest number of need for coloring. Denoted as . Lower bound Theorem: For every deterministic online algorithm there exists a logn-colorable graph for which the algorithm uses at least 2n/logn colors. The performance ratio of any deterministic online coloring algorithm is at least . Transparent online coloring game Adversary strategy : The collection of all subsets of {1,2,...,k} of size k/2. Avail(vt): Admissible colors consists of colors not used by its pre-neighbors. Hue(b)={Corlor(vi): Bin(vi) = b}: hue of a bin is the set of colors of vertices in the bin. H: hue collection is a set of all nonempty hues. #bin >= n/(k/2) #color<=k ratio>=2n/(k*k) Lower bound Theorem: For every randomized online algorithm there exists a k- colorable graph on which the algorithm uses at least n/k bins, where k=O(logn). The performance ratio of any randomized online coloring algorithm is at least . Adversary strategy for randomized algo Relaxing the constraint - blocked input Theorem: The performance ratio of any randomized algorithm, when the input is presented in blocks of size , is . Relaxing other constraints 1. Look-ahead and bufferring 2. Recoloring 3. Presorting vertices by degree 4. Disclosing the adversary’s previous coloring First Fit Use the smallest numbered color that does not violate the coloring requirement Induced subgraph A induced subgraph is a subset of the vertices of a graph G together with any edges whose endpoints are both in the subset. -
Hyperbolicity and Complement of Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 24 (2011) 1882–1887 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Hyperbolicity and complement of graphs Sergio Bermudo a, José M. Rodríguez b, José M. Sigarreta c,∗, Eva Tourís b a Departamento de Economía, Métodos cuantitativos e Historia Económica, Universidad Pablo de Olavide, Carretera de Utrera Km. 1, 41013 Sevilla, Spain b Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain c Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, 39650 Acalpulco Gro., Mexico article info a b s t r a c t Article history: If X is a geodesic metric space and x1; x2; x3 2 X, a geodesic triangle T D fx1; x2; x3g is the Received 20 July 2010 union of the three geodesics Tx1x2U, Tx2x3U and Tx3x1U in X. The space X is δ-hyperbolic (in Received in revised form 29 April 2011 the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two Accepted 11 May 2011 other sides, for every geodesic triangle T in X. We denote by δ.X/ the sharp hyperbolicity constant of X, i.e. δ.X/ VD inffδ ≥ 0 V X is δ-hyperbolicg. The study of hyperbolic graphs is Keywords: an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the Graph hyperbolicity of a graph related to it. -
Induced Path Factors of Regular Graphs Arxiv:1809.04394V3 [Math.CO] 1
Induced path factors of regular graphs Saieed Akbari∗ Daniel Horsley† Ian M. Wanless† Abstract An induced path factor of a graph G is a set of induced paths in G with the property that every vertex of G is in exactly one of the paths. The induced path number ρ(G) of G is the minimum number of paths in an induced path factor of G. We show that if G is a connected cubic graph on n > 6 vertices, then ρ(G) 6 (n − 1)=3. Fix an integer k > 3. For each n, define Mn to be the maximum value of ρ(G) over all connected k-regular graphs G on n vertices. As n ! 1 with nk even, we show that ck = lim(Mn=n) exists. We prove that 5=18 6 c3 6 1=3 and 3=7 6 c4 6 1=2 and that 1 1 ck = 2 − O(k− ) for k ! 1. Keywords: Induced path, path factor, covering, regular graph, subcubic graph. Classifications: 05C70, 05C38. 1 Introduction We denote the path of order n by Pn. A subgraph H of a graph G is said to be induced if, for any two vertices x and y of H, x and y are adjacent in H if and only if they are adjacent in G. An induced path factor (IPF) of a graph G is a set of induced paths in G with the property that every vertex of G is in exactly one of the paths. We allow paths of any length in an IPF, including the trivial path P1. -
Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width∗†
Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width∗† Lars Jaffke‡1, O-joung Kwon§2, and Jan Arne Telle3 1 Department of Informatics, University of Bergen, Norway [email protected] 2 Logic and Semantics, Technische Universität Berlin, Berlin, Germany [email protected] 3 Department of Informatics, University of Bergen, Norway [email protected] Abstract We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give nO(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H-Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Per- mutation and Circular Permutation graphs, Convex graphs, k-Trapezoid, Circular k-Trapezoid, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, Computations on discrete structures, G.2.2 Graph Theory, Graph algorithms Keywords and phrases graph width parameters, dynamic programming, graph classes, induced paths, induced topological minors Digital Object Identifier 10.4230/LIPIcs.IPEC.2017.21 1 Introduction Ever since the definition of the tree-width of graphs emerged from the Graph Minors project of Robertson and Seymour, bounded-width structural graph decompositions have been a successful tool in designing fast algorithms for graph classes on which the corresponding width-measure is small. -
Long Induced Paths in Graphs∗ Arxiv:1602.06836V2 [Math.CO] 1
Long induced paths in graphs∗ Louis Esperety Laetitia Lemoinez Fr´ed´ericMaffrayx December 2, 2016 Abstract We prove that every 3-connected planar graph on n vertices contains an induced path on Ω(log n) vertices, which is best possible and improves the best known lower bound by a multiplicative factor of log log n. We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) with a path on n vertices, also contains an induced path on Ω(plog n) vertices. We conjecture that for any k, there is a positive constant c(k) such that any k-degenerate graph with a path on n vertices also contains an induced path on Ω((log n)c(k)) vertices. We provide examples showing that this order of magnitude would be best possible (already for chordal graphs), and prove the conjecture in the case of interval graphs. 1 Introduction A graph contains a long induced path (i.e., a long path as an induced subgraph) only if it contains a long path. However, this necessary condition is not sufficient, as shown by complete graphs and complete bipartite graphs. On the other hand, it was proved by Atminas, Lozin and Ragzon [2] that if a graph G contains a long path, but does not contain a large complete graph or complete bipartite graph, then G contains a long induced path. Their proof uses several applications of Ramsey theory, and the resulting bound on the size of a long induced path is thus quantitatively weak. The specific case of k-degenerate graphs (graphs such that any subgraph contains a arXiv:1602.06836v2 [math.CO] 1 Dec 2016 vertex of degree at most k) was considered by Neˇsetˇriland Ossona de Mendez in [6]. -
An Introduction to Algebraic Graph Theory
An Introduction to Algebraic Graph Theory Cesar O. Aguilar Department of Mathematics State University of New York at Geneseo Last Update: March 25, 2021 Contents 1 Graphs 1 1.1 What is a graph? ......................... 1 1.1.1 Exercises .......................... 3 1.2 The rudiments of graph theory .................. 4 1.2.1 Exercises .......................... 10 1.3 Permutations ........................... 13 1.3.1 Exercises .......................... 19 1.4 Graph isomorphisms ....................... 21 1.4.1 Exercises .......................... 30 1.5 Special graphs and graph operations .............. 32 1.5.1 Exercises .......................... 37 1.6 Trees ................................ 41 1.6.1 Exercises .......................... 45 2 The Adjacency Matrix 47 2.1 The Adjacency Matrix ...................... 48 2.1.1 Exercises .......................... 53 2.2 The coefficients and roots of a polynomial ........... 55 2.2.1 Exercises .......................... 62 2.3 The characteristic polynomial and spectrum of a graph .... 63 2.3.1 Exercises .......................... 70 2.4 Cospectral graphs ......................... 73 2.4.1 Exercises .......................... 84 3 2.5 Bipartite Graphs ......................... 84 3 Graph Colorings 89 3.1 The basics ............................. 89 3.2 Bounds on the chromatic number ................ 91 3.3 The Chromatic Polynomial .................... 98 3.3.1 Exercises ..........................108 4 Laplacian Matrices 111 4.1 The Laplacian and Signless Laplacian Matrices .........111 4.1.1 -
Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width∗
MULTISCALE MODEL. SIMUL. c xxxx Society for Industrial and Applied Mathematics Vol. xx, No. x, pp. x–x ALMOST OPTIMAL LOWER BOUNDS FOR PROBLEMS PARAMETERIZED BY CLIQUE-WIDTH∗ FEDOR V. FOMIN† , PETR A. GOLOVACH† , DANIEL LOKSHTANOV† , AND SAKET SAURABH‡ Abstract. We obtain asymptotically tight algorithmic bounds for Max-Cut and Edge Dom- inating Set problems on graphs of bounded clique-width. We show that on an n-vertex graph of clique-width t both problems • cannot be solved in time f(t)no(t) for any function f of t unless Exponential Time Hy- pothesis (ETH) fails, and • can be solved in time nO(t). Key words. Exponential Time Hypothesis, clique-width, max-cut, edge dominating set AMS subject classifications. 05C85, 68R10, 68Q17, 68Q25, 68W40 1. Introduction. Tree-width is one of the most fundamental parameters in graph algorithms. Graphs of bounded tree-width enjoy good algorithmic properties similar to trees and this is why many problems which are hard on general graphs can be solved efficiently when the input is restricted to graphs of bounded tree-width. On the other hand, many hard problems also become tractable when restricted to graphs “similar to complete graphs”. Courcelle and Olariu [6] introduced the notion of clique-width which captures nice algorithmic properties of both extremes. Since 2000, the research on algorithmic and structural aspects of clique-width is an active direction in Graph Algorithms, Logic, and Complexity. Corneil et al. [4] show that graphs of clique-width at most 3 can be recognized in polynomial time. Fellows et al. [12] settled a long standing open problem by showing that computing clique-width is NP-hard.