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Results on Domination Number and Bondage Number for Some Families of Graphs
International Journal of Pure and Applied Mathematics Volume 107 No. 3 2016, 565-577 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v107i3.5 ijpam.eu RESULTS ON DOMINATION NUMBER AND BONDAGE NUMBER FOR SOME FAMILIES OF GRAPHS G. Hemalatha1, P. Jeyanthi2 § 1Department of Mathematics Shri Andal Alagar College of Engineering Mamandur, Kancheepuram, Tamil Nadu, INDIA 2Research Centre, Department of Mathematics Govindammal Aditanar College for Women Tiruchendur 628 215, Tamil Nadu, INDIA Abstract: Let G = (V, E) be a simple graph on the vertex set V . In a graph G, A set S ⊆ V is a dominating set of G if every vertex in VS is adjacent to some vertex in S. The domination number of a graph Gγ(G)] is the minimum size of a dominating set of vertices in G. The bondage number of a graph G[Bdγ(G)] is the cardinality of a smallest set of edges whose removal results in a graph with domination number larger than that of G. In this paper we establish domination number and the bondage number for some families of graphs. AMS Subject Classification: 05C69 Key Words: dominating set, bondage number, domination number, cocktail party graph, coxeter graph, crown graph, cubic symmetric graph, doyle graph, folkman graph, levi graph, icosahedral graph 1. Introduction The domination in graphs is one of the major areas in graph theory which attracts many researchers because it has the potential to solve many real life problems involving design and analysis of communication network, social net- Received: October 17, 2015 c 2016 Academic Publications, Ltd. -
Instituto De Computação ✄✂ ✁✂✄✁✂✄✁✂ ✄✁ ✄✂✁ ✄✂ ✁✄✂ ✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁ Universidade Estadual De Campinas
✄✄ ✄ ✄ ✄ ✄✄ ✄✂ ✁✂✄✁✂ ✄✁✂ ✄✁ ✄✂ ✁✄ ✂✄✁✂ ✁ ✄✂ ✄✁ ✄✂ ✄✁✂ ✁✂✄✁✂ ✁✄✂ ✁✄✂ ✁ INSTITUTO DE COMPUTAÇÃO ✄✂ ✁✂✄✁✂✄ ✁✂✄ ✁✄ ✂ ✁✄✂ ✁✄✂ ✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁ UNIVERSIDADE ESTADUAL DE CAMPINAS Colorings and crossings Atílio G. Luiz R. Bruce Richter Technical Report - IC-13-18 - Relatório Técnico August - 2013 - Agosto The contents of this report are the sole responsibility of the authors. O conteúdo do presente relatório é de única responsabilidade dos autores. Colorings and crossings∗ At´ılioG. Luiz† R. Bruce Richter‡ Abstract In 2007, Albertson conjectured that if a graph G has chromatic number k, then the crossing number of G is at least the crossing number of the complete graph with k vertices. To date, two papers were written on this subject trying to solve the conjecture for an arbitrary k-chromatic graph G, and after much effort the conjecture was proved true for k ≤ 16. In this report, we present an overview of topics related to Albertson’s Conjecture, such as: the crossing number problem, lower bounds on crossing number, lower bounds on the number of edges of color-critical graphs, and coloring of graphs with small crossing number and small clique number. While investigating Albertson’s conjecture, J. Bar´at and G. T´oth proposed the following conjecture involving color- critical graphs: for every positive integer c, there exists a bound k(c) such that for any k, where k ≥ k(c), any k-critical graph on k + c vertices has a subdivision of Kk. In Section 9, we present counterexamples to this conjecture for every c ≥ 6 and we prove that the conjecture is valid for c = 5. 1 Introduction A graph G is planar if it can be drawn in the plane so that its edges intersect only at their ends. -
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. -
Prime Labeling of Union Graphs
International Journal of Scientific Research in _____________________________ Research Paper . Mathematical and Statistical Sciences Vol.6, Issue.1, pp.86-93, February (2019) E-ISSN: 2348-4519 DOI: https://doi.org/10.26438/ijsrmss/v6i1.8693 Prime labeling of Union Graphs S K Patel1, A N Kansagara2* 1 Department of Mathematics, Government Engineering College, Bhuj-370 001. India. 2 Department of Mathematics, Shree D K V Arts and Science College, Jamnagar-361 008. India. *Corresponding author: [email protected] Tel: +91 94267 57504 Available online at: www.isroset.org Received: 01/Feb/2019, Accepted: 11/Feb/2019, Online: 28/Feb/2019 Abstract- A graph with vertices is called prime graph if its vertices can be labeled with first positive integers such that each pair of adjacent vertices have relatively prime labels. Such a labeling is called prime labeling. It is known that graphs like gear graphs, crown graphs, helm graphs, book graphs and are prime. In this paper, we study the prime labeling of union of these graphs. Keyword: Prime labeling, union of graphs, Corona product, Cartesian Product. AMS Subject Classification(2010): 05C7 I. INTRODUCTION (iii) whereas and are odd integers (iv) , where is any integer. All graphs considered in this paper are simple, finite, and 2. undirected. For a graph , and denotes the 3. , where is an odd integer. vertex set and edge set of respectively whereas 4. For arbitrary integers if and denotes the cardinality of the respective sets. For , then . terms not defined here, we refer the reader to [6]. Paths and cycles are the immediate examples of Now we discuss some basic definions which will be used in prime graphs but on the other hand the conjecture that trees main results. -
Smith Forms for Adjacency Matrices of Circulant Graphs
Smith forms for adjacency matrices of circulant graphs Gerald Williamsa aDepartment of Mathematical Sciences, University of Essex, Colchester, Essex CO4 3SQ, U.K. Abstract We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andr´asfaigraph, Doob graph, cocktail party graph, crown graph, prism graph, M¨obiusladder. The proofs operate by finding the abelianisation of a cyclically presented group whose relation matrix is column equivalent to the required adjacency matrix. Keywords: Smith normal form, circulant graph, adjacency matrix. 2000 MSC: 05C50, 15A21, 20F05. 1. Introduction The circulant matrix circn(a0; : : : ; an−1) is the n × n matrix whose first row is (a0; : : : ; an−1) and where row (i+1) (0 ≤ i ≤ n−2) is a cyclic shift of row i by one column. A circulant graph is a graph that is isomorphic to a graph whose adjacency matrix is circulant. We shall write A(Γ) for the adjacency matrix of a graph Γ. Given graphs Γ; Γ0 if det(A(Γ)) =6 det(A(Γ0)) (in particular if precisely one of A(Γ), A(Γ0) is singular) then Γ; Γ0 are non-isomorphic. Similarly, if rank(A(Γ)) =6 rank(A(Γ0)) then Γ; Γ0 are non-isomorphic. Singularity, rank, and determinants of various families of circulant graphs are considered (for example) in [3],[7],[16],[23]. For an n × n integer matrix M, the Smith normal form of M, written SNF(M), is the n × n diagonal integer matrix S = diagn(d0; : : : ; dn−1) where d0; : : : ; dn−1 2 N [ f0g and dijdi+1 (0 ≤ i ≤ n − 2) is such that there exist invertible integer matrices P; Q such that PMQ = S. -
Pattern Avoidance and Fiber Bundle Structures on Schubert Varieties
Thursday 2.10 Auditorium B Pattern Avoidance and Fiber Bundle Structures on Schubert Varieties Timothy Alland [email protected] SUNY Stony Brook (This talk is based on joint work with Edward Richmond.) MSC2000: 05E15, 14M15 Permutations are known to index type A Schubert varieties. A question one can ask regarding a specific Schubert variety is whether it is an iterated fiber bundle of Grass- mannian Schubert varieties, that is, if it has a complete parabolic bundle structure. We have found that a Schubert variety has a complete parabolic bundle structure if and only if its associated permutation avoids the patterns 3412, 52341, and 635241. We find this by first identifying when the standard projection from the Schubert variety in the complete flag variety to the Schubert variety in the Grassmannian is a fiber bundle using what we have called \split pattern avoidance". In this talk, I will demonstrate how we were able to move from a characterization of this projection in terms of the support and left descents of the permutation's parabolic decomposition to one that applies split pattern avoidance. I will also give a flavor of the proof of how the three patterns mentioned above determine whether a Schubert variety has a complete parabolic bundle structure. Monday 3.55 Auditorium B A simple proof of Shamir's conjecture Peter Allen [email protected] LSE (This talk is based on joint work with Julia B¨ottcher, Ewan Davies, Matthew Jenssen, Yoshiharu Kohayakawa, Barnaby Roberts.) MSC2000: 05C80 It is well known (and easy to show) that the threshold for a perfect matching in the bino- log n mial random graph G(n; p) is p = Θ n , coinciding with the threshold for every vertex to be in an edge (and much more is known). -
Arxiv:2108.05223V2 [Math.CO] 12 Aug 2021 the Line Graph of the Crown
The line graph of the crown graph is distance integral S. Morteza Mirafzal Department of Mathematics Lorestan University, Khorramabad, Iran E-mail: [email protected] E-mail: [email protected] Abstract The distance eigenvalues of a connected graph G are the eigenvalues of its distance matrix D(G). A graph is called distance integral if all of its distance eigenvalues are integers. Let n ≥ 3 be an integer. A crown graph Cr(n) is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. Let L(Cr(n)) denote the line graph of the crown graph Cr(n). In this paper, by using the orbit partition method in algebraic graph theory, we determine the set of all distance eigenvalues of L(Cr(n)) and show that this graph is distance integral. 1 Introduction and Preliminaries arXiv:2108.05223v4 [math.CO] 8 Sep 2021 In this paper, a graph G = (V, E) is considered as an undirected simple graph where V = V (G) is the vertex-set and E = E(G) is the edge-set. For all the terminology and notation not defined here, we follow [3,4,5,6,7]. Let G = (V, E) be a graph and A = A(G) be an adjacency matrix of G. The characteristic polynomial of G is defined as P (G; x)= P (x)= |xI − A|. A zero of p(x) is called an eigenvalue of the graph G. A graph is called integral, if all the eigenvalues are integers. The study of integral graphs was initiated by 2010 Mathematics Subject Classification:05C50 Keywords: crown graph, distance integral, vertex-transitive, orbit partition Date: 1 2 S. -
Crossings, Colorings, and Cliques
Crossings, colorings, and cliques Michael O. Albertson ∗ Daniel W. Cranston † Jacob Fox ‡ Abstract Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds by Pach et al. on the crossing number of graphs in terms of the number of edges and vertices. 1 Introduction For more than a century, from Kempe through Appel and Haken and continuing to the present, the Four Color Problem [5, 31] has played a leading role in the development of graph theory. For background we recommend the classic book by Jensen and Toft [18]. There are three classic relaxations of planarity. The first is that of a graph embedded on an arbitrary surface. Here Heawood established an upper bound for the number of colors needed to color any embedded graph. About forty years ago Ringel and Youngs completed the work of showing that the Heawood bound is (with the exception of Klein’s bottle) sharp. Shortly thereafter Appel and Haken proved the Four Color Theorem. One consequence of these results is that the maximum chromatic number of a graph embedded on any given surface is achieved by a complete graph. -
Context-Bounded Analysis of Concurrent Queue Systems*
Context-Bounded Analysis of Concurrent Queue Systems Salvatore La Torre1, P. Madhusudan2, and Gennaro Parlato1,2 1 Universit`a degli Studi di Salerno, Italy 2 University of Illinois at Urbana-Champaign, USA Abstract. We show that the bounded context-switching reachability problem for concurrent finite systems communicating using unbounded FIFO queues is decidable, where in each context a process reads from only one queue (but is allowed to write onto all other queues). Our result also holds when individual processes are finite-state recursive programs provided a process dequeues messages only when its local stack is empty. We then proceed to classify architectures that admit a decidable (un- bounded context switching) reachability problem, using the decidability of bounded context switching. We show that the precise class of decid- able architectures for recursive programs are the forest architectures, while the decidable architectures for non-recursive programs are those that do not have an undirected cycle. 1 Introduction Networks of concurrent processes communicating via message queues form a very natural and useful model for several classes of systems with inherent paral- lelism. Two natural classes of systems can be modeled using such a framework: asynchronous programs on a multi-core computer and distributed programs com- municating on a network. In parallel programming languages for multi-core or even single-processor sys- tems (e.g., Java, web service design), asynchronous programming or event-driven programming is a common idiom that programming languages provide [19,7,13]. In order to obtain higher performance and low latency, programs are equipped with the ability to issue tasks using asynchronous calls that immediately return, but are processed later, either in parallel with the calling module or perhaps much later, depending on when processors and other resources such as I/O be- come free. -
On the Parameterized Complexity of Polytree Learning
On the Parameterized Complexity of Polytree Learning Niels Grüttemeier , Christian Komusiewicz and Nils Morawietz∗ Philipps-Universität Marburg, Marburg, Germany {niegru, komusiewicz, morawietz}@informatik.uni-marburg.de Abstract In light of the hardness of handling general Bayesian networks, the learning and inference prob- A Bayesian network is a directed acyclic graph that lems for Bayesian networks fulfilling some spe- represents statistical dependencies between vari- cific structural constraints have been studied exten- ables of a joint probability distribution. A funda- sively [Pearl, 1989; Korhonen and Parviainen, 2013; mental task in data science is to learn a Bayesian Korhonen and Parviainen, 2015; network from observed data. POLYTREE LEARN- Grüttemeier and Komusiewicz, 2020a; ING is the problem of learning an optimal Bayesian Ramaswamy and Szeider, 2021]. network that fulfills the additional property that its underlying undirected graph is a forest. In One of the earliest special cases that has received atten- this work, we revisit the complexity of POLYTREE tion are tree networks, also called branchings. A tree is a LEARNING. We show that POLYTREE LEARN- Bayesian network where every vertex has at most one par- ING can be solved in 3n · |I|O(1) time where n is ent. In other words, every connected component is a di- the number of variables and |I| is the total in- rected in-tree. Learning and inference can be performed in stance size. Moreover, we consider the influence polynomial time on trees [Chow and Liu, 1968; Pearl, 1989]. of the number of variables d that might receive a Trees are, however, very limited in their modeling power nonempty parent set in the final DAG on the com- since every variable may depend only on at most one other plexity of POLYTREE LEARNING. -
Mathematisches Forschungsinstitut Oberwolfach Graph Theory
Mathematisches Forschungsinstitut Oberwolfach Report No. 2/2016 DOI: 10.4171/OWR/2016/2 Graph Theory Organised by Reinhard Diestel, Hamburg Daniel Kr´al’, Warwick Paul Seymour, Princeton 10 January – 16 January 2016 Abstract. This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results. Mathematics Subject Classification (2010): 05C. Introduction by the Organisers The aim of the workshop was to offer a forum to communicate recent develop- ments in graph theory and discuss directions for further research in the area. The atmosphere of the workshop was extremely lively and collaborative. On the first day of the workshop, each participant introduced her/himself and briefly presented her/his research interests, which helped to establish a working atmosphere right from the very beginning of the workshop. The workshop program consisted of 9 long more general talks and 21 short more focused talks. The talks were comple- mented by six evening workshops to discuss particular topics at a larger depth; these were held in ad hoc formed groups in the evenings. The workshop focused on four interlinked topics in graph theory, which have recently seen new exciting developments. These topics were nowhere-zero flows and the dual notion of graph colorings, • sparsity of graphs and its algorithmic applications, • width parameters and graph decompositions, and • results following the proof of Rota’s conjecture in matroid theory. • 52 Oberwolfach Report 2/2016 The four main themes of the workshop were reflected in the selection of the work- shop participants and in the choice of the talks for the program. -
Graph Theory Graph Theory (II)
J.A. Bondy U.S.R. Murty Graph Theory (II) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.