DOCSLIB.ORG

Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-Connected Graphs with Bounded Maximum Degrees

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-Connected Graphs with Bounded Maximum Degrees Georgia State University ScholarWorks @ Georgia State University Mathematics Dissertations Department of Mathematics and Statistics 5-11-2015 Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees Songling Shan Follow this and additional works at: https://scholarworks.gsu.edu/math_diss Recommended Citation Shan, Songling, "Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees." Dissertation, Georgia State University, 2015. https://scholarworks.gsu.edu/math_diss/23 This Dissertation is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Dissertations by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees by Songling Shan Under the Direction of Guantao Chen, PhD ABSTRACT A tree T with no vertex of degree 2 is called a homeomorphically irreducible tree (HIT) and if T is spanning in a graph, then T is called a homeomorphically irreducible spanning tree (HIST). Albertson, Berman, Hutchinson and Thomassen asked if every triangulation of at least 4 vertices has a HIST and if every connected graph with each edge in at least two triangles contains a HIST. These two questions were restated as two conjectures by Archdeacon in 2009. The first part of this dissertation gives a proof for each of the two conjectures. The second part focuses on some problems about Halin graphs, which is a class of graphs closely related to HITs and HISTs. A Halin graph is obtained from a plane embedding of a HIT of at least 4 vertices by connecting its leaves into a cycle following the cyclic order determined by the embedding. And a generalized Halin graph is obtained from a HIT of at least 4 vertices by connecting the leaves into a cycle. Let G be a sufficiently large n-vertex graph. Applying the Regularity Lemma and the Blow-up Lemma, it is shown that G contains a spanning Halin subgraph if it has minimum degree at least (n +1)/2 and G contains a spanning generalized Halin subgraph if it is 3-connected and has minimum degree at least (2n + 3)/5. The minimum degree conditions are best possible. The last part estimates the length of longest cycles in 3-connected graphs with bounded maximum degrees. In 1993 Jackson and Wormald conjectured that for any positive integer d 4, there ≥ exists a positive real number α depending only on d such that if G is a 3-connected n-vertex graph with maximum degree d, then G has a cycle of length at least αnlogd−1 2. They showed that the exponent in the bound is best possible if the conjecture is true. The conjecture is confirmed for d 425. ≥ INDEX WORDS: Homeomorphically irreducible spanning tree, Halin graph, Genaralized Halin graph, 3-connected graphs, Tutte decomposition. Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees by Songling Shan A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the College of Arts and Sciences Georgia State University 2015 Copyright by Songling Shan 2015 Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees by Songling Shan Committee Chair: Guantao Chen Committee: Ron Gould Frank Hall Zhongshan Li Hendricus Van der Holst Yi Zhao Electronic Version Approved: Office of Graduate Studies College of Arts and Sciences Georgia State University May 2015 iv DEDICATION This dissertation is dedicated to my family and my advisor Professor Guantao Chen. v ACKNOWLEDGEMENTS This dissertation work would not have been possible without the support of many people. First and foremost, I would like to express my sincere gratitude to my advisor Professor Guantao Chen. He serves not just as an academic advisor, but also like a friend and a father. What I appreciate most is his always-open office door, whenever I have questions, I can just walk in, talk to him and discuss with him. His guidance and suggestions keep me going forward for tracking research problems and living positively. On Chinese new year and other Chinese holidays, he invites us to his home for dinner. For students like me who are thousands of miles away from home, he offers us the home-like comfort. There are so many words of gratitude that I want to express, but I just don’t know where and how to start. Above all, it is the biggest treasure in my life to have Professor Chen as my advisor. Thanks also to Professor Frank Hall and Professor Zhongshan Li. Their help and encouragement lightened my life at GSU. I wish to thank the other members of my dissertation committee: Professors Ron Gould, Hendricus Van der Holst and Yi Zhao. In addition, I give my gratitude to all the other faculty and staff members in the department. Also, I would like to thank my group members Nana Li, Ping Yang, and Amy Yates. Beyond metropolitan Atlanta, I sincerely thank Professor Mark Ellingham of Vanderbilt University, whom I got to know in my first year at GSU, and who has helped me as a mentor; I also sincerely thank Professor Akira Saito, he is like my secondary advisor. Further back, I would like to thank my first graph theory enlightenment teacher Professor Bing Yao. It was he who helped me in writing the first research paper; and that was the very primary motivation of my stepping on the research journey in graph theory. Then it is the time to say thanks to my Master of Science advisor Professor Han Ren, and one of my favorite professors at East China Normal University, Professor Xingzhi Zhan. Finally and most importantly, I thank my family members both in China and in the US for their love and support. Thinking of them, deep in my heart, can always calm me down, and is always a wonderful thing to do. vi TABLE OF CONTENTS ACKNOWLEDGEMENTS ................. v LIST OF FIGURES .................... viii LIST OF ABBREVIATIONS ................ ix PART 1 INTRODUCTION ............... 1 PART 2 THE EXISTENCE OF HISTS IN SURFACE TRIANGU- LATIONS AND CONNECTED GRAPHS WITH EACH EDGE IN AT LEAST TWO TRIANGLES ...... 3 2.1 Proof of Conjecture 2.1 .......................... 4 2.1.1 ProofofTheorem2.1.1........................ 4 2.2 Proof of Conjecture 2.2 .......................... 7 2.2.1 ProofofTheorem2.2.1........................ 8 PART 3 MINIMUM DEGREECONDITION FOR SPANNINGHALIN GRAPHS AND SPANNINGGENERALIZED HALIN GRAPHS 16 3.1 Notations and definitions ......................... 16 3.2 The Regularity Lemma and the Blow-up Lemma .......... 17 3.3 Dirac’s sondition for spanning Halin graphs ............. 18 3.3.1 Introduction.............................. 18 3.3.2 Ladders and “ladder-like” Halin graphs . 21 3.3.3 ProofofTheorem3.3.1........................ 23 3.3.3.1 Proof of Theorem 3.3.2 . 26 3.3.3.2 Proof of Theorem 3.3.3 . 30 3.3.3.3 Proof of Theorem 3.4.3 . 38 vii 3.4 Minimum degree condition for spanning generalized Halin graphs 48 3.4.1 Introduction.............................. 48 3.4.2 Proof of Theorem 3.4.1 and the sharpness of Theorem 3.4.2 . 50 3.4.3 ProofofTheorem3.4.2........................ 52 3.4.3.1 Proof of Theorem 3.4.3 . 53 3.4.3.2 Proof of Theorem 3.4.4 . 75 3.4.3.3 Proof of Theorem 3.4.5 . 84 PART 4 A LOWER BOUND ON CIRCUMFERENCES OF 3- CONNECTED GRAPHS WITH BOUNDED MAXIMUM DEGREES .................. 102 4.1 Introduction ................................. 102 4.2 Paths in block-chains ........................... 104 4.3 Lower bounds of mlogb 2 + nlogb 2 ...................... 108 4.4 Long paths in block-chains ........................ 111 4.5 Proofs of Theorem (4.1.1) (a) and (b) ................. 126 4.5.1 Case 1: t 2 or t 2. ...................... 129 1 ≥ 2 ≥ 4.5.2 Case 2. t1 = t2 =1. ......................... 131 4.6 Reduction of Theorem (4.1.1)(c) .................... 133 4.6.1 Case 1 x / p, q . .......................... 140 ∈{ } 4.6.1.1 Subcase1.1. p, q = a , b = V (H H ). ..... 141 { } 6 { t t} t ∩ t+1 4.6.1.2 Case p, q = a , b = V (H H ).......... 143 { } { t t} t ∩ t+1 4.6.2 Case 2 x p, q . .......................... 149 ∈{ } REFERENCES ..................... 167 viii LIST OF FIGURES Figure2.1 G4, a locally connected graph without a spanning 2-tree . 5 Figure2.2 Θ-graph,∆-operation,Θ-operation . .. 9 Figure2.3 uw / E(T ′) or uw E(T ′) E(G) ................ 13 1 ∈ 1 ∈ ∩ Figure2.4 uw E(T ′) E(G)......................... 15 1 ∈ − Figure3.1 L4, Hi constructed from L4, and Ti associated with Hi for each i = 1, 2, , 5 ............................... 24 · · · Figure3.2 Ladder L oforder14......................... 26 Figure3.3 A Halin graph H ........................... 29 Figure3.4 The HIT TB .............................. 64 Figure3.5 The tree TA .............................. 66 Figure3.6 The tree T11 .............................. 77 Figure3.7 T with I =1............................ 92 W | | Figure3.8 F with M =3........................... 93 M | | ix LIST OF ABBREVIATIONS GSU - Georgia State University • HIT - Homeomorphically irreducible tree • HIST - Homeomorphically irreducible spanning tree • SGHG - Spanning generalized Halin subgraph • 1 PART 1 INTRODUCTION Finding longest cycles, in particular a hamiltonian cycle in a graph, is one of a few fundamental yet very difficult problems in graph theory. In fact, to determine whether a graph is hamiltonian is a classic NP-complete problem. Moreover, Karger, Motwani, and Ramkumar [35] showed that, unless = , it is impossible to find, in polynomial time, a P N P path of length n nǫ in an n-vertex hamiltonian graph for any ǫ < 1.
Load more

Recommended publications
  • Zadání Bakalářské Práce
    ČESKÉ VYSOKÉ UČENÍ TECHNICKÉ V PRAZE FAKULTA INFORMAČNÍCH TECHNOLOGIÍ ZADÁNÍ BAKALÁŘSKÉ PRÁCE Název: Rekurzivně konstruovatelné grafy Student: Anežka Štěpánková Vedoucí: RNDr. Jiřina Scholtzová, Ph.D. Studijní program: Informatika Studijní obor: Teoretická informatika Katedra: Katedra teoretické informatiky Platnost zadání: Do konce letního semestru 2016/17 Pokyny pro vypracování Souvislé rekurentně zadané grafy jsou souvislé grafy, pro které existuje postup, jak ze zadaného grafu odvodit graf nový. V předmětu BI-GRA jste se s jednoduchými rekurentními grafy setkali, vždy šly popsat lineárními rekurentními rovnicemi s konstantními koeficienty. Existuje klasifikace, která definuje třídu rekurzivně konstruovatelných grafů [1]. Podobně jako u grafů v BI-GRA zjistěte, zda lze některé z kategorií rekurzivně konstruovatelných grafů popsat rekurentními rovnicemi. 1. Seznamte se s rekurentně zadanými grafy. 2. Seznamte se s rekurzivně konstruovatelnými grafy a nastudujte články o nich. 3. Vyberte si některé z kategorií rekurzivně konstruovatelných grafů [1]. 4. Pro vybrané kategorie najděte způsob, jak je popsat využitím rekurentních rovnic. Seznam odborné literatury [1] Gross, Jonathan L., Jay Yellen, Ping Zhang. Handbook of Graph Theory, 2nd ed, Boca Raton: CRC, 2013. ISBN: 978-1-4398-8018-0 doc. Ing. Jan Janoušek, Ph.D. prof. Ing. Pavel Tvrdík, CSc. vedoucí katedry děkan V Praze dne 18. února 2016 České vysoké učení technické v Praze Fakulta informačních technologií Katedra Teoretické informatiky Bakalářská práce Rekurzivně konstruovatelné grafy Anežka Štěpánková Vedoucí práce: RNDr. Jiřina Scholtzová Ph.D. 12. května 2017 Poděkování Děkuji vedoucí práce RNDr. Jiřině Scholtzové Ph.D., která mi ochotně vyšla vstříc s volbou tématu a v průběhu tvorby práce vždy dobře poradila a na- směrovala mne správným směrem.
    [Show full text]
  • Minor-Closed Graph Classes with Bounded Layered Pathwidth
    Minor-Closed Graph Classes with Bounded Layered Pathwidth Vida Dujmovi´c z David Eppstein y Gwena¨elJoret x Pat Morin ∗ David R. Wood { 19th October 2018; revised 4th June 2020 Abstract We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class. 1 Introduction Pathwidth and treewidth are graph parameters that respectively measure how similar a given graph is to a path or a tree. These parameters are of fundamental importance in structural graph theory, especially in Roberston and Seymour's graph minors series. They also have numerous applications in algorithmic graph theory. Indeed, many NP-complete problems are solvable in polynomial time on graphs of bounded treewidth [23]. Recently, Dujmovi´c,Morin, and Wood [19] introduced the notion of layered treewidth. Loosely speaking, a graph has bounded layered treewidth if it has a tree decomposition and a layering such that each bag of the tree decomposition contains a bounded number of vertices in each layer (defined formally below). This definition is interesting since several natural graph classes, such as planar graphs, that have unbounded treewidth have bounded layered treewidth. Bannister, Devanny, Dujmovi´c,Eppstein, and Wood [1] introduced layered pathwidth, which is analogous to layered treewidth where the tree decomposition is arXiv:1810.08314v2 [math.CO] 4 Jun 2020 required to be a path decomposition.
    [Show full text]
  • A Note on Max-Leaves Spanning Tree Problem in Halin Graphs
    AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 29 (2004), Pages 95–97 A note on max-leaves spanning tree problem in Halin graphs Dingjun Lou Huiquan Zhu Department of Computer Science Zhongshan University Guangzhou 510275 People’s Republic of China Abstract A Halin graph H is a planar graph obtained by drawing a tree T in the plane, where T has no vertex of degree 2, then drawing a cycle C through all leaves in the plane. We write H = T ∪ C,whereT is called the characteristic tree and C is called the accompanying cycle. The problem is to find a spanning tree with the maximum number of leaves in a Halin graph. In this paper, we prove that the characteristic tree in a Halin graph H is one of the spanning trees with maximum number of leaves in H, and we design a linear time algorithm to find such a tree in the Halin graph. All graphs in this paper are finite, undirected and simple. We follow the termi- nology and notation of [1] and [6]. A Halin graph H is a planar graph obtained by drawing a tree T in the plane, where T has no vertex of degree 2, then drawing a cycle through all leaves of T in the plane. We write H = T ∪ C,whereT is called the characteristic tree and C is called the accompanying cycle. In this paper, we deal with the problem of finding a spanning tree with maximum number of leaves in a Halin graph. The problem of finding a spanning tree with maximum number of leaves for arbitrary graphs is NP-complete (see [3], Problem [ND 2]).
    [Show full text]
  • Lombardi Drawings of Graphs 1 Introduction
    Lombardi Drawings of Graphs Christian A. Duncan1, David Eppstein2, Michael T. Goodrich2, Stephen G. Kobourov3, and Martin Nollenburg¨ 2 1Department of Computer Science, Louisiana Tech. Univ., Ruston, Louisiana, USA 2Department of Computer Science, University of California, Irvine, California, USA 3Department of Computer Science, University of Arizona, Tucson, Arizona, USA Abstract. We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. 1 Introduction The American artist Mark Lombardi [24] was famous for his drawings of social net- works representing conspiracy theories. Lombardi used curved arcs to represent edges, leading to a strong aesthetic quality and high readability. Inspired by this work, we intro- duce the notion of a Lombardi drawing of a graph, in which edges are drawn as circular arcs with perfect angular resolution: consecutive edges are evenly spaced around each vertex. While not all vertices have perfect angular resolution in Lombardi’s work, the even spacing of edges around vertices is clearly one of his aesthetic criteria; see Fig. 1. Traditional graph drawing methods rarely guarantee perfect angular resolution, but poor edge distribution can nevertheless lead to unreadable drawings. Additionally, while some tools provide options to draw edges as curves, most rely on straight-line edges, and it is known that maintaining good angular resolution can result in exponential draw- ing area for straight-line drawings of planar graphs [17,25].
    [Show full text]
  • Outerplanar Graphs
    MSOL-definability equals recognizability for Halin graphs and bounded degree $k$-outerplanar graphs Citation for published version (APA): Jaffke, L., & Bodlaender, H. L. (2015). MSOL-definability equals recognizability for Halin graphs and bounded degree $k$-outerplanar graphs. (arXiv; Vol. 1503.01604 [cs.LO]). s.n. Document status and date: Published: 01/01/2015 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
    [Show full text]
  • The Strong Chromatic Index of Cubic Halin Graphs
    The Strong Chromatic Index of Cubic Halin Graphs Ko-Wei Lih∗ Daphne Der-Fen Liu Institute of Mathematics Department of Mathematics Academia Sinica California State University, Los Angeles Taipei 10617, Taiwan Los Angeles, CA 90032, U. S. A. Email:[email protected] Email:[email protected] April 19, 2011 Abstract A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if they are incident to a common edge or share an endpoint. The strong chromatic index of a graph G, denoted sχ0(G), is the minimum number of colors needed for a strong edge coloring of G. A Halin graph G is a plane graph constructed from a tree T without vertices of degree two by connecting all leaves through a cycle C. If a cubic Halin graph G is different 0 from two particular graphs Ne2 and Ne4, then we prove sχ (G) 6 7. This solves a conjecture proposed in W. C. Shiu and W. K. Tam, The strong chromatic index of complete cubic Halin graphs, Appl. Math. Lett. 22 (2009) 754{758. Keywords: Strong edge coloring; Strong chromatic index; Halin graph. 1 Introduction For a graph G with vertex set V (G) and edge set E(G), the line graph L(G) of G is the graph on the vertex set E(G) such that two vertices in L(G) are defined to be adjacent if and only if their corresponding edges in G share a common endpoint. The distance between two edges in G is defined to be their distance in L(G).
    [Show full text]
  • Some Combinatorial Problems on Halin Graphs
    Some Combinatorial Problems on Halin Graphs M.Kavin 1, K.Keerthana 1, N.Sadagopan 2, Sangeetha.S 1, and R.Vinothini 1 1 Department of Computer Science and Engineering, College of Engineering Guindy, Chennai, India. 2 Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, India. [email protected] Abstract. Let T be a tree with no degree 2 vertices and L(T ) = {l1,...,lr},r ≥ 2 denote the set of leaves in T . An Halin graph G is a graph obtained from T such that V (G) = V (T ) and E(G) = E(T ) ∪ {{li,li+1} | 1 ≤ i ≤ r − 1} ∪ {l1,lr}. In this paper, we investigate combinatorial problems such as, testing whether a given graph is Halin or not, chromatic bounds, an algorithm to color Halin graphs with the minimum number of colors. Further, we present polynomial-time algorithms for testing and coloring problems. 1 Introduction The study of special graph classes like bipartite graphs, chordal graphs, planar graphs have attracted the re- searchers both from theoretical and application perspective. On the theoretical front, combinatorial problems like vertex cover, odd cycle transversal which are otherwise NP-complete in general graphs have polynomial- time algorithms restricted to chordal graphs [1]. However, problems like Hamilton path, odd cycle transversal are still NP-complete even on special graph like planar graphs [2] which calls for identifying a non-trivial subclass of planar graphs where Hamilton path (cycle) is polynomial-time solvable. One such subclass is the class of Halin graphs. Halin graph is a graph obtained from a tree with no degree two vertices by joining all leaves with a cycle [5] and by construction Halin graphs are planar.
    [Show full text]
  • Solving Problems on Recursively Constructed Graphs
    Solving Problems on Recursively Constructed Graphs RICHARD B. BORIE University of Alabama and R. GARY PARKER and CRAIG A. TOVEY Georgia Institute of Technology Fast algorithms can be created for many graph problems when instances are confined to classes of graphs that are recursively constructed. This paper first describes some basic conceptual notions regarding the design of such fast algorithms, and then the coverage proceeds through several recursive graph classes. Specific classes include k-terminal graphs, trees, series-parallel graphs, k-trees, partial k-trees, Halin graphs, bandwidth-k graphs, pathwidth-k graphs, treewidth- k graphs, branchwidth-k graphs, cographs, cliquewidth-k graphs, k-NLC graphs, k-HB graphs, and rankwidth-k graphs. The definition of each class is provided, after which some typical algorithms are applied to solve problems on instances of each class. Categories and Subject Descriptors: G.2.2 [Discrete Mathematics]: Graph Theory; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Prob- lems General Terms: Algorithms, Theory Additional Key Words and Phrases: Bandwidth, branchwidth, cliquewidth, cograph, Halin graph, pathwidth, rankwidth, series parallel, tree, treewidth 1. INTRODUCTION Throughout, G =(V,E) denotes a finite graph where V and E are sets of vertices and edges respectively. A recursively constructed graph class is defined by a set (usually finite) of primitive or base graphs, in addition to one or more operations (called composition rules) that compose larger graphs from smaller subgraphs. Each operation involves fusing specific vertices from each subgraph or adding new edges between specific vertices from each subgraph. Each graph in a recursive class has a corresponding decomposition tree that reveals how to build it from base graphs.
    [Show full text]
  • The Vulcan Game of Kal-Toh: Finding Or Making Triconnected Planar Subgraphs
    The Vulcan game of Kal-toh: Finding or making triconnected planar subgraphs by Terry David Anderson A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2011 c Terry David Anderson 2011 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In the game of Kal-toh depicted in the television series Star Trek: Voyager, players attempt to create polyhedra by adding to a jumbled collection of metal rods. Inspired by this fictional game, we formulate graph-theoretical questions about polyhedral (triconnected and planar) subgraphs in an on-line environment. The problem of determining the existence of a polyhedral subgraph within a graph G is shown to be NP-hard, and we also give some non-trivial upper bounds for the problem of determining the minimum number of edge additions necessary to guarantee the existence of a polyhedral subgraph in G. A two-player formulation of Kal-toh is also explored, in which the first player to form a target subgraph is declared the winner. We show a polynomial-time solution for simple cases of this game but conjecture that the general problem is NP-hard. iii Acknowledgements I am truly grateful for the unending support, encouragement, and patience of my super- visor, Therese Biedl. I would also like to thank my thesis readers, Craig Kaplan and Alex L´opez-Ortiz, for their feedback and suggestions.
    [Show full text]
  • 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.
    [Show full text]
  • Minimum Degree Condition for Spanning Generalized Halin Graphs
    Minimum degree condition for spanning generalized Halin graphs Guantao Chen, Songling Shan, and Ping Yang Georgia State University, Atlanta, GA 30303, USA Abstract. A spanning tree with no vertices of degree 2 is called a Homeomorphically irreducible spanning tree (HIST). Based on a HIST embedded in the plane, a Halin graph is formed by connect- ing the leaves of the tree into a cycle following the cyclic order determined by the embedding. Both of the determination problems of whether a graph contains a HIST or whether a graph contains a spanning Halin graph are shown to be NP-complete. It was conjectured by Albertson, Berman, Hutchinson, and Thomassen in 1990 that a every surface triangulation of at least four vertices con- tains a HIST (confirmed). And it was conjectured by Lov´asz and Plummer that every 4-connected plane triangulation contains a spanning Halin graph (disproved). Balancing the above two facts, in this paper, we consider generalized Halin graphs, a family of graph structures which are “stronger” than HISTs but “weaker” than Halin graphs in the sense of their construction constraints. To be exact, a generalized Halin graph is formed from a HIST by connecting its leaves into a cycle. Since a generalized Halin graph needs not to be planar, we investigate the minimum degree condition for a graph to contain it as a spanning subgraph. We show that there exists a positive integer n0 such that any 3-connected graph with n ≥ n0 vertices and minimum degree at least (2n + 3)/5 contains a spanning generalized Halin graph. As an application, the result implies that under the same condition, the graph G contains a wheel-minor of order at least n/2.
    [Show full text]
  • Listing All Spanning Trees in Halin Graphs-Sequential and Parallel View
    Listing All Spanning Trees in Halin Graphs - Sequential and Parallel view K.Krishna Mohan Reddy, P.Renjith, N.Sadagopan Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, India. {coe11b014,coe14d002,sadagopan}@iiitdm.ac.in Abstract. For a connected labelled graph G, a spanning tree T is a connected and an acyclic subgraph that spans all vertices of G. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of G. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of O((2pd)p) processors for parallel algorithmics, where d and p are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is O((2pd)p). 1 Introduction Enumeration of sets satisfying a specific property is an important combinatorial problem in the field of combinatorics and computing. Popular ones are listing all spanning trees [1,2,3,4], listing all minimal ver- tex separators [5], enumerating maximal independent sets [6], etc. In [7], G.J.Minty initiated the study of enumeration of spanning trees in general graphs as it finds applications in circuits and network systems. Subsequently, Read et al. [8] and Shioura et al. [9] presented an algorithm to list all spanning trees in ar- bitrary graphs. So far, algorithms for enumeration of spanning trees reported in the literature either follow backtracking approach or enumeration with the help of fundamental cycles.
    [Show full text]