Linear Separation of Connected Dominating Sets in Graphs (Extended Abstract) Nina Chiarelli1 and Martin Milanicˇ2
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:2106.16130V1 [Math.CO] 30 Jun 2021 in the Special Case of Cyclohedra, and by Cardinal, Langerman and P´Erez-Lantero [5] in the Special Case of Tree Associahedra
LAGOS 2021 Bounds on the Diameter of Graph Associahedra Jean Cardinal1;4 Universit´elibre de Bruxelles (ULB) Lionel Pournin2;4 Mario Valencia-Pabon3;4 LIPN, Universit´eSorbonne Paris Nord Abstract Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of search trees on G, defined recursively by a root r together with search trees on each of the connected components of G − r. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n − 6 for a large enough n. We give a tight bound of Θ(m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n). Keywords: generalized permutohedra, graph associahedra, tree-depth, treewidth, pathwidth 1 Introduction The vertices and edges of a polyhedron form a graph whose diameter (often referred to as the diameter of the polyhedron for short) is related to a number of computational problems. For instance, the question of how large the diameter of a polyhedron arises naturally from the study of linear programming and the simplex algorithm (see, for instance [27] and references therein). -
On Computing Longest Paths in Small Graph Classes
On Computing Longest Paths in Small Graph Classes Ryuhei Uehara∗ Yushi Uno† July 28, 2005 Abstract The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for weighted trees, block graphs, and cacti. We next show that the longest path problem can be solved efficiently on some graph classes that have natural interval representations. Keywords: efficient algorithms, graph classes, longest path problem. 1 Introduction The Hamiltonian path problem is one of the most well known NP-hard problem, and there are numerous applications of the problems [17]. For such an intractable problem, there are two major approaches; approximation algorithms [20, 2, 35] and algorithms with parameterized complexity analyses [15]. In both approaches, we have to change the decision problem to the optimization problem. Therefore the longest path problem is one of the basic problems from the viewpoint of combinatorial optimization. From the practical point of view, it is also very natural approach to try to find a longest path in a given graph, even if it does not have a Hamiltonian path. However, finding a longest path seems to be more difficult than determining whether the given graph has a Hamiltonian path or not. -
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. -
The Hadwiger Number, Chordal Graphs and Ab-Perfection Arxiv
The Hadwiger number, chordal graphs and ab-perfection∗ Christian Rubio-Montiel [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onomade M´exico, 04510, Mexico City, Mexico Department of Algebra, Comenius University, 84248, Bratislava, Slovakia October 2, 2018 Abstract A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph H, (1) the Hadwiger number of H is equal to the maximum clique order of H, (2) the Hadwiger number of H is equal to the achromatic number of H, (3) the b-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-b-chromatic number is equal to the pseudoachromatic number, (5) the arXiv:1701.08417v1 [math.CO] 29 Jan 2017 Hadwiger number of H is equal to the Grundy number of H, and (6) the b-chromatic number is equal to the pseudo-Grundy number. Keywords: Complete colorings, perfect graphs, forbidden graphs characterization. 2010 Mathematics Subject Classification: 05C17; 05C15; 05C83. ∗Research partially supported by CONACyT-Mexico, Grants 178395, 166306; PAPIIT-Mexico, Grant IN104915; a Postdoctoral fellowship of CONACyT-Mexico; and the National scholarship programme of the Slovak republic. 1 1 Introduction Let G be a finite graph. A k-coloring of G is a surjective function & that assigns a number from the set [k] := 1; : : : ; k to each vertex of G.A k-coloring & of G is called proper if any two adjacent verticesf haveg different colors, and & is called complete if for each pair of different colors i; j [k] there exists an edge xy E(G) such that x &−1(i) and y &−1(j). -
Ssp Structure of Some Graph Classes
International Journal of Pure and Applied Mathematics Volume 101 No. 6 2015, 939-948 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: A http://www.ijpam.eu P ijpam.eu SSP STRUCTURE OF SOME GRAPH CLASSES R. Mary Jeya Jothi1, A. Amutha2 1,2Department of Mathematics Sathyabama University Chennai, 119, INDIA Abstract: A Graph G is Super Strongly Perfect if every induced sub graph H of G possesses a minimal dominating set that meets all maximal cliques of H. Strongly perfect, complete bipartite graph, etc., are some of the most important classes of super strongly perfect graphs. Here, we analyse the other classes of super strongly perfect graphs like trivially perfect graphs and very strongly perfect graphs. We investigate the structure of super strongly perfect graph in friendship graphs. Also, we discuss the maximal cliques, colourability and dominating sets in friendship graphs. AMS Subject Classification: 05C75 Key Words: super strongly perfect graph, trivially perfect graph, very strongly perfect graph and friendship graph 1. Introduction Super strongly perfect graph is defined by B.D. Acharya and its characterization has been given as an open problem in 2006 [5]. Many analysation on super strongly perfect graph have been done by us in our previous research work, (i.e.,) we have investigated the classes of super strongly perfect graphs like strongly c 2015 Academic Publications, Ltd. Received: March 12, 2015 url: www.acadpubl.eu 940 R.M.J. Jothi, A. Amutha perfect, complete bipartite graph, etc., [3, 4]. In this paper we have discussed some other classes of super strongly perfect graphs like trivially perfect graph, very strongly perfect graph and friendship graph. -
Introducing Subclasses of Basic Chordal Graphs
Introducing subclasses of basic chordal graphs Pablo De Caria 1 Marisa Gutierrez 2 CONICET/ Universidad Nacional de La Plata, Argentina Abstract Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we find some other properties they have, mostly involving clique graphs. Keywords: Chordal graph, dually chordal graph, clique tree, compatible tree. 1 Introduction 1.1 Definitions For a graph G, V (G) is the set of its vertices and E(G) is the set of its edges. A clique is a maximal set of pairwise adjacent vertices. The family of cliques of G is denoted by C(G). For v ∈ V (G), Cv will denote the family of cliques containing v. All graphs considered will be assumed to be connected. A set S ⊆ V (G) is a uv-separator if vertices u and v are in different connected components of G−S. It is minimal if no proper subset of S has the 1 Email: [email protected] aaURL: www.mate.unlp.edu.ar/~pdecaria 2 Email: [email protected] same property. S is a minimal vertex separator if there exist two nonadjacent vertices u and v such that S is a minimal uv-separator. S(G) will denote the family of all minimal vertex separators of G. -
Graph Layout for Applications in Compiler Construction
Theoretical Computer Science Theoretical Computer Science 2 I7 ( 1999) 175-2 I4 Graph layout for applications in compiler construction G. Sander * Abstract We address graph visualization from the viewpoint of compiler construction. Most data struc- tures in compilers are large, dense graphs such as annotated control flow graph, syntax trees. dependency graphs. Our main focus is the animation and interactive exploration of these graphs. Fast layout heuristics and powerful browsing methods are needed. We give a survey of layout heuristics for general directed and undirected graphs and present the browsing facilities that help to manage large structured graphs. @ l999-Elsevier Science B.V. All rights reserved Keyrror& Graph layout; Graph drawing; Compiler visualization: Force-directed placement; Hierarchical layout: Compound drawing; Fisheye view 1. Introduction We address graph visualization from the viewpoint of compiler construction. Draw- ings of compiler data structures such as syntax trees, control flow graphs, dependency graphs [63], are used for demonstration, debugging and documentation of compilers. In real-world compiler applications, such drawings cannot be produced manually because the graphs are automatically generated, large, often dense, and seldom planar. Graph layout algorithms help to produce drawings automatically: they calculate positions of nodes and edges of the graph in the plane. Our main focus is the animation and interactive exploration of compiler graphs. Thus. fast layout algorithms are required. Animations show the behaviour of an algorithm by a running sequence of drawings. Thus there is not much time to calculate a layout between two subsequent drawings. In interactive exploration, it is annoying if the user has to wait a long time for a layout. -
Dominating Cliques in 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) 101-116 101 North-Holland DOMINATING CLIQUES IN GRAPHS Margaret B. COZZENS Mathematics Department, Northeastern University, Boston, MA 02115, USA Laura L. KELLEHER Massachusetts Maritime Academy, USA Received 2 December 1988 A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding a dominating clique. A forbidden subgraph characterization is given for a class of graphs that have a connected dominating set of size three. Introduction A set of vertices in a simple, undirected graph is a dominating set if every vertex in the graph which is not in the dominating set is adjacent to one or more vertices in the dominating set. The domination number of a graph G is the minimum number of vertices in a dominating set. Several types of dominating sets have been investigated, including independent dominating sets, total dominating sets, and connected dominating sets, each of which has a corresponding domination number. Dominating sets have applications in a variety of fields, including communication theory and political science. For more background on dominating sets see [3, 5, 151 and other articles in this issue. For arbitrary graphs, the problem of finding the size of a minimum dominating set in the graph is an NP-complete problem [9]. -
On Dasgupta's Hierarchical Clustering Objective and Its Relation to Other
On Dasgupta's hierarchical clustering objective and its relation to other graph parameters Svein Høgemo1, Benjamin Bergougnoux1, Ulrik Brandes3, Christophe Paul2, and Jan Arne Telle1 1 Department of Informatics, University of Bergen, Norway 2 LIRMM, CNRS, Univ Montpellier, France 3 Social Networks Lab, ETH Z¨urich, Switzerland Abstract. The minimum height of vertex and edge partition trees are well-studied graph parameters known as, for instance, vertex and edge ranking number. While they are NP-hard to determine in general, linear- time algorithms exist for trees. Motivated by a correspondence with Das- gupta's objective for hierarchical clustering we consider the total rather than maximum depth of vertices as an alternative objective for mini- mization. For vertex partition trees this leads to a new parameter with a natural interpretation as a measure of robustness against vertex removal. As tools for the study of this family of parameters we show that they have similar recursive expressions and prove a binary tree rotation lemma. The new parameter is related to trivially perfect graph completion and there- fore intractable like the other three are known to be. We give polynomial- time algorithms for both total-depth variants on caterpillars and on trees with a bounded number of leaf neighbors. For general trees, we obtain a 2-approximation algorithm. 1 Introduction Clustering is a central problem in data mining and statistics. Although many objective functions have been proposed for (flat) partitions into clusters, hier- archical clustering has long been considered from the perspective of iterated merge (in agglomerative clustering) or split (in divisive clustering) operations. -
PDF of the Phd Thesis
Durham E-Theses Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs THOMAS, DANIEL,JAMES,RHYS How to cite: THOMAS, DANIEL,JAMES,RHYS (2020) Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/13671/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas A Thesis presented for the degree of Doctor of Philosophy Department of Computer Science Durham University United Kingdom June 2020 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas Submitted for the degree of Doctor of Philosophy June 2020 Abstract: In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdős, Ko and Rado to graphs. -
The Bidimensionality Theory and Its Algorithmic
The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi B.S., Sharif University of Technology, 2000 M.S., University of Waterloo, 2001 Submitted to the Department of Mathematics in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 °c MohammadTaghi Hajiaghayi, 2005. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Mathematics April 29, 2005 Certi¯ed by. Erik D. Demaine Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Rodolfo Ruben Rosales Chairman, Applied Mathematics Committee Accepted by . Pavel I. Etingof Chairman, Department Committee on Graduate Students 2 The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi Submitted to the Department of Mathematics on April 29, 2005, in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY Abstract Our newly developing theory of bidimensional graph problems provides general techniques for designing e±cient ¯xed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k £ k grid graph (and similar graphs) grows with k, typically as (k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). -
Algorithms for Minimum M-Connected K-Tuple Dominating Set Problem$
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 381 (2007) 241–247 www.elsevier.com/locate/tcs Algorithms for minimum m-connected k-tuple dominating set problem$ Weiping Shanga,c,∗, Pengjun Wanb, Frances Yaoc, Xiaodong Hua a Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 10080, China b Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA c Department of Computer Science, City University of Hong Kong, Hong Kong Received 24 January 2007; received in revised form 25 April 2007; accepted 29 April 2007 Communicated by D.-Z. Du Abstract In wireless sensor networks, a virtual backbone has been proposed as the routing infrastructure to alleviate the broadcasting storm problem and perform some other tasks such as area monitoring. Previous work in this area has mainly focused on how to set up a small virtual backbone for high efficiency, which is modelled as the minimum Connected Dominating Set (CDS) problem. In this paper we consider how to establish a small virtual backbone to balance efficiency and fault tolerance. This problem can be formalized as the minimum m-connected k-tuple dominating set problem, which is a general version of minimum CDS problem with m = 1 and k = 1. We propose three centralized algorithms with small approximation ratios for small m and improve the current best results for small k. c 2007 Elsevier B.V. All rights reserved. Keywords: Connected dominating set; Approximation algorithm; k-vertex connectivity; Wireless sensor networks 1. Introduction A Wireless Sensor Network (WSN) consists of wireless nodes (transceivers) without any underlying physical infrastructure.