DOCSLIB.ORG

Degree Sequences, Forcibly Chordal Graphs, and Combinatorial Proof Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Degree Sequences, Forcibly Chordal Graphs, and Combinatorial Proof Systems DEGREE SEQUENCES, FORCIBLY CHORDAL GRAPHS, AND COMBINATORIAL PROOF SYSTEMS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Christian Altomare, B.S. Graduate Program in Mathematics The Ohio State University 2009 Dissertation Committee: Dr. G. Neil Robertson, Advisor Dr. John Maharry Dr. Akos Seress ABSTRACT We study the structure theory of two combinatorial objects closely related to graphs. First, we consider degree sequences, and we prove several results originally moti- vated by attempts to prove what was, until recently, S.B. Rao's conjecture, and what is now a theorem of Paul Seymour and Maria Chudnovsky, namely, that graphic degree sequences are well quasi ordered. We give a new, surprisingly non-graph the- oretic proof of the bounded case of this theorem. Next, we obtain an exact structure theorem of degree sequences excluding a square and a pentagon. Using this result, we then prove a structure theorem for degree sequences excluding a square and, more generally, excluding an arbitrary cycle. It should be noted that taking complements, this yields a structure theorem for excluding a matching. The structure theorems above, however, are stated in terms of forcibly chordal graphs, hence we next begin their characterization. While an exact characterization seems difficult, certain partial results are obtained. Specifically, we first characterize the degree sequences of forcibly chordal trees. Next, we use this result to extend the characterization to forcibly chordal forests. Finally, we characterize forcibly chordal graphs having a certain path structure. Next, we define a class of combinatorial objects that generalizes digraphs and partial orders, which is motivated by proof systems arising in mathematical logic. We ii give what we believe will be the basic theory of these objects, including definitions, theorems, and proofs. We define the minors of a proof system, and we give two forbidden minors theorems, one of them characterizing partial orders as proof systems by forbidden minors. iii To Moomar. iv ACKNOWLEDGMENTS First and foremost, I wish to thank Neil Robertson, my advisor. It is every student's wish to have an advisor with such depth of understanding, breadth of knowledge, and raw intuition for his field of expertise. I have gained from him not only knowledge, but an understanding of how research mathematics is carried out. His ability to find the right generalization to prove, the right special case to consider, the right approach to try, and the right question to ask at all, has continually amazed me. Second, I would like to thank S.B. Rao for a beautiful conjecture. Third, I would like to thank Christopher McClain for his generous and patient help related to typesetting and document preparation, which are not my strong suits. Fourth, I wish to thank Akos Seress and John Maharry for their time and effort participating in my thesis committee. Fifth, I would like to thank everyone in the Ohio State University Mathematics Department who has helped me in my time since I started taking mathematics courses here as a high school student. In particular, in the order in which I met them, I am thankful to John Maharry, Alexander Dynin, Judie Monson, Yung-Chen Lu, Vitaly Bergelson, Randall Dougherty, Tim Carlson, Cindy Bernlohr, Boris Pittel, and once again my advisor for the amount of time, effort, and patience they were willing to spend toward my career and development. I thank the countless others in the v department who have helped me as well. Without their help, this would not be possible. Sixth, I thank my parents, Richard and Karen Altomare. vi VITA April 7, 1980 . Born - Columbus, OH 1998-2001 . Undergraduate, The Ohio State University 2001 . B.S. in Mathematics, The Ohio State University 2001-Present . Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major Field: Mathematics Specialization: Graph Theory vii TABLE OF CONTENTS Abstract . ii Dedication . iv Acknowledgments . v Vita . vii CHAPTER PAGE 1 Introduction . 1 1.1 Introduction to Degree Sequences . 1 1.2 Degree Sequence Basics, Notation, and Conventions . 5 1.3 Introduction to Combinatorial Proof Systems . 9 2 The Bounded Case of Rao's Conjecture . 12 3 Excluding Matchings and Cycles . 24 4 Forcibly Chordal Trees . 42 5 Forcibly Chordal Forests . 54 6 Forcibly Chordal Graphs . 59 7 Combinatorial Proof Systems . 86 7.1 Introduction . 86 7.2 Proof Closure . 87 7.3 The Merge . 88 7.4 Preceding Set Proof Systems . 92 7.4.1 Examples . 94 7.4.2 Motivation for Definition of Proof . 96 7.4.3 Proof Definition and Basics . 97 viii 7.4.4 Autonomous Sets . 99 7.4.5 Axioms . 103 7.4.6 The Information in the Set of Proofs . 104 7.5 Autonomous Systems . 106 7.5.1 Introduction . 106 7.5.2 Definition of Proof Revisited . 107 7.6 The Canonical Orders . 107 7.6.1 Canonical Order Definition and Basics . 107 7.6.2 Descendability . 109 7.6.3 Canonical Order Basic Theorems And Examples . 116 7.7 Partial Orders As Ausyses . 120 7.8 Well Founded Autonomous Systems . 128 7.9 Blocking . 133 7.9.1 The Blocking Order . 140 7.9.2 Blocking In Posets . 143 7.10 Ausys Lexicographic Sum . 144 7.11 Subausys, Dot, Homomorphisms, and Minors . 150 7.12 Relations to Matching and Connectivity . 163 Bibliography . 165 ix CHAPTER 1 INTRODUCTION This work studies two classes of objects. The first class we study is the class of degree sequences of finite graphs. The second class we study is a class of combinatorial proof systems we call autonomous systems. 1.1 Introduction to Degree Sequences We assume familiarity with basic graph theory. Definitions and conventions are as in [3] unless otherwise stated. Our graphs are finite, simple, and undirected throughout unless otherwise stated. Definition 1.1.1. Let G be a graph with vertices v1; : : : ; vn, listed such that d(v1) ≥ · · · ≥ d(vn). Then the degree sequence of G, denoted by D(G), is the sequence (d(v1); : : : ; d(vn)). We make no use of the fact that, according to our definition, the degree sequence is a decreasing sequence. It is rather simply the easiest way to make the degree sequence of a graph unique, so we can refer to the degree sequence D(G) of G, as opposed to a degree sequence of G. We note that while a degree sequence does not technically have any vertices, it can be 1 very suggestive to think of the vertices of a degree sequence, which we sometimes do. The degree sequence (2; 2; 2; 1; 1), for instance, would be said to have three vertices of degree 2 and two of degree 1. Definition 1.1.2. Let D be a degree sequence and let G be a graph. We say that G realizes D, or that G is a realization of D, if D(G) = D. We denote by R(D) the set of realizations of D. Myriad theorems in combinatorics, and in particular graph theory, study the graphs not \containing" a fixed graph, for various notions of containment. It is fruitful to define a notion of containment for degree sequences as well so that similar questions may be asked and theorems proved. Definition 1.1.3. Let D1 and D2 be degree sequences. We write D1 ≤ D2 if there is a graph G1 in R(D1) and a graph G2 in R(D2) such that G1 is an induced subgraph of G2. The reader may check that ≤ is a reflexive, transitive relation. One motivation for making this definition is that the induced subgraph relation for graphs can be extremely difficult to work with, even for questions that are tractable if the induced subgraph relation is replaced with another containment relation. The relation ≤ for degree sequences is similar to, but more tractable in many cases than, the induced subgraph relation for graphs. 2 A discussion of claw free graphs and degree sequences best illustrates this point. A claw is the unique graph up to isomorphism with degree sequence (3; 1; 1; 1). Suppose we wish to find the structure of claw free graphs. What claw free means of course depends on the containment relation used. If we work with the minor relation, we are asking which graphs have no claw as a minor. It is trivial that a graph is claw free in this sense iff it has no vertices of degree three or more. The claw free graphs are trivially then exactly the disjoint unions of paths and cycles. If instead of working with the minor relation, we rather work with the induced sub- graph relation, the structure of claw free graphs is then a deep and difficult theorem of Chudnovsky and Seymour, proved in a series of five papers totalling over 200 pages. Now, if instead of working with graphs excluding a claw as an induced subgraph, we instead ask which degree sequences exclude the degree sequence of a claw, then the structure theorem given by Robertson and Song can be proved in under six pages. Thus, in passing from induced subgraphs to the ≤ relation on degree sequences, we have a theorem that is motivated by induced subgraphs, yet still more amenable to analysis. With this motivation, degree sequence analogues of questions asked for graphs are of- ten asked for degree sequences. The celebrated Minor Theorem of Robertson and Sey- mour says that finite graphs are well quasi ordered under the minor relation. A well quasi order is a reflexive, transitive relation T on a set X such that if x1; x2; : : : ; xn;::: is an infinite sequence in X then there exist i and j with i < j such that xiT xj.
Load more

Recommended publications
  • Graph Partitions and the Bichromatic Number Dennis D.A. Epple Doctor of Philosophy
    Graph Partitions and the Bichromatic Number by Dennis D.A. Epple Diplom, Freie Universit¨atBerlin, 2005 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics and Statistics c Dennis D.A. Epple, 2011 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part by photocopy or other means, without the permission of the author. Graph Partitions and the Bichromatic Number by Dennis D.A. Epple Diplom, Freie Universit¨atBerlin, 2005 Supervisory Committee Dr. Jing Huang, Supervisor (Department of Mathematics and Statistics) Dr. Peter Dukes, Member (Department of Mathematics and Statistics) Dr. Gary MacGillivray, Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) ii Supervisory Committee Dr. Jing Huang, Supervisor (Department of Mathematics and Statistics) Dr. Peter Dukes, Member (Department of Mathematics and Statistics) Dr. Gary MacGillivray, Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) Abstract A(k; l)-colouring of a graph is a partition of its vertex set into k independent sets and l cliques. The bichromatic number χb of a graph is the minimum r such that the graph is (k; l)-colourable for all k + l = r. The bichromatic number is related to the cochromatic number, which can also be defined in terms of (k; l)-colourings. The bichromatic number is a fairly recent graph parameter that arises in the study of extremal graphs related to a classical result of Erd}os,Stone and Simonovits, and in the study of the edit distance of graphs from hereditary graph classes.
    [Show full text]
  • Families of Integral Cographs Within a Triangular Array Received July 3, 2020; Accepted November 8, 2020
    Spec. Matrices 2020; 8:257–273 Research Article Open Access Hsin-Yun Ching, Rigoberto Flórez*, and Antara Mukherjee Families of Integral Cographs within a Triangular Array https://doi.org/10.1515/spma-2020-0116 Received July 3, 2020; accepted November 8, 2020 Abstract: The determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three innite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sucient condition for a graph from these families to be integral. Some features of these graphs are: they are integral cographs, all graphs have at most ve distinct eigenval- ues, all graphs are either d-regular graphs with d = 2, 4, 6, ... or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle. Keywords: Fibonacci number, determinant Hosoya triangle, adjacency matrix, eigenvalue, integral graph, cograph MSC: Primary 68R11; Secondary 11B39 1 Introduction A graph is integral if the eigenvalues of its adjacency matrix are integers. These graphs are rare and the tech- niques used to nd them are quite complicated. The notion of integral graphs was rst introduced in 1974 by Harary and Schwenk [17]. A cograph is a graph that avoids the 4 vertices path as an induced subgraph [11]. In this paper we study an innite family of integral cographs associated with a combinatorial triangle. These families have at most ve distinct eigenvalues.
    [Show full text]
  • Modular Decomposition of Undirected Graphs
    THESIS MODULAR DECOMPOSITION OF UNDIRECTED GRAPHS Submitted by Adrian E. Esparza Department of Computer Science In partial fulfillment of the requirements For the Degree of Master of Science Colorado State University Fort Collins, Colorado Summer 2020 Master’s Committee: Advisor: Ross McConnell Sangmi Pallickara Alexander Hulpke Copyright by Adrian E. Esparza 2020 All Rights Reserved ABSTRACT MODULAR DECOMPOSITION OF UNDIRECTED GRAPHS Graphs found in nature tend to have structure and interesting compositions that allow for com- pression of the information and thoughtful analysis of the relationships between vertices. Modular decomposition has been studied since 1967 [1]. Modular decomposition breaks a graph down into smaller components that, from the outside, are inseparable. In doing so, modules provide great po- tential to better study problems from genetics to compression. This paper describes the implemen- tation of a practical algorithm to take a graph and decompose it into it modules in O(n+m log(n)) time. In the implementation of this algorithm, each sub-problem was solved using object ori- ented design principles to allow a reader to extract the individual objects and turn them to other problems of practical interest. The purpose of this paper is to provide the reader with the tools to easily compute: modular decomposition of an undirected graph, partition an undirected graph, depth first search on a directed partially complemented graph, and stack operations with a com- plement stack. The provided implementation of these problems all compute within the time bound discussed above, or even faster for several of the sub problems. ii ACKNOWLEDGEMENTS I would like to thank my family for their ceaseless support.
    [Show full text]
  • [Math.LO] 18 Apr 2004 on the Order of Countable Graphs
    On the Order of Countable Graphs Jaroslav Neˇsetˇril∗ Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI) Charles University Malostransk´en´am.25, 11800 Praha 1 Czech Republic Email: [email protected]ff.cuni.cz Saharon Shelah Hebrew University, Jerusalem† Institute of Mathematics The Hebrew University of Jerusalem Jerusalem 91904, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08854, USA Email: [email protected] October 7, 2018 arXiv:math/0404319v1 [math.LO] 18 Apr 2004 1 ∗Partially supported by the Project LN00A056 of the Czech Ministry of Education and by GAUK 158 grant. †Research supported by the United States-Israel Binational Science Foundation. Publication 745 1Math. Subj. Class.03C, 05C, 05E, and 06A. Keywords and phrases: Density, Partially ordered sets, Rigid graphs, Universal graphs. The authors wish to thank the Mittag-Leffler Institute, for its support and pleasant environment during September, 2000, when this paper was written. 1 Abstract A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2ω the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended (“partnership theorem”). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. This is related to universal graphs, rigid graphs (where we solve a problem posed in [19]) and to the density problem for countable graphs.
    [Show full text]
  • University of Bergen Maximum Number of Edges in Graphs Under Various Constraints
    University of Bergen Department of Informatics Algorithms Maximum Number of Edges in Graphs Under Various Constraints Student: H˚avard Stuberg Supervisor: Pinar Heggernes Masters Thesis June, 2019 ii iii Acknowledgements I would like to thank my supervisor Pinar Heggernes for all the help, guidance and encouragement I have received while working on this thesis. I also want to thank Paloma Lima for her help and suggestions. iv v Contents 1 Introduction1 1.1 Notation and Definitions..........................1 1.2 Graph Classes................................3 1.2.1 Chordal Graphs...........................3 1.2.2 Interval Graphs...........................4 1.2.3 Split Graphs.............................5 1.2.4 Trivially Perfect Graphs......................5 1.2.5 Cographs..............................6 1.2.6 Graph Class Hierarchy.......................7 1.3 Overview of the Thesis...........................8 2 Bounding the Number of Edges in a Graph9 2.1 Maximum Degree and Matching Number................. 12 2.2 Maximum Degree and Induced Matching Number............ 16 3 Bounded Matching Number and Maximum Degree 19 3.1 Chordal Graphs............................... 20 3.1.1 Cliques and Stars.......................... 20 3.1.2 Test Results............................. 21 3.2 Trivially Perfect Graphs.......................... 23 3.3 Interval Graphs............................... 24 4 Bounded Induced Matching Number and Maximum Degree 29 4.1 Chordal Graphs............................... 30 vi CONTENTS 4.2 Interval Graphs............................... 32 4.3 Split Graphs................................. 33 4.4 Cographs.................................. 35 5 Conclusion 37 5.1 The Practical Framework for Testing Hypotheses............ 37 5.1.1 Generating Graphs......................... 38 5.1.2 Testing the Graphs......................... 40 5.2 A Related Problem: Minimal Feedback Vertex Sets in Chordal Graphs 41 5.2.1 Maximum Number of Minimal Feedback Vertex Sets in Chordal Graphs...............................
    [Show full text]
  • A Vizing-Like Theorem for Union Vertex-Distinguishing Edge Coloring Nicolas Bousquet, Antoine Dailly, Eric Duchene, Hamamache Kheddouci, Aline Parreau
    A Vizing-like theorem for union vertex-distinguishing edge coloring Nicolas Bousquet, Antoine Dailly, Eric Duchene, Hamamache Kheddouci, Aline Parreau To cite this version: Nicolas Bousquet, Antoine Dailly, Eric Duchene, Hamamache Kheddouci, Aline Parreau. A Vizing- like theorem for union vertex-distinguishing edge coloring. Discrete Applied Mathematics, Elsevier, 2017, 232, pp.88-98. 10.1016/j.dam.2017.07.002. hal-01313088v2 HAL Id: hal-01313088 https://hal.archives-ouvertes.fr/hal-01313088v2 Submitted on 17 Jul 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Vizing-like theorem for union vertex-distinguishing edge coloring Nicolas Bousqueta, Antoine Daillyb,∗, Eric´ Duch^eneb, Hamamache Kheddoucib, Aline Parreaub aG-SCOP (CNRS, Univ. Grenoble-Alpes), Grenoble, France. bUniv Lyon, Universit´eLyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France. Abstract We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the problem of finding a coloring with the minimum number of colors where every vertex receives a distinct label.
    [Show full text]
  • Local Recognition of Reflection Graphs on Coxeter Groups
    Local recognition of reflection graphs on Coxeter groups Diploma Thesis 31 Mar 2008 (minor corrections, 13 May 2008) Armin Straub Technische Universität Darmstadt Department of Mathematics Algebra, Geometry, Functional Analysis supervised by PD dr. Ralf Gramlich Contents 3 Contents Contents .............................................. 3 Notations .............................................. 4 1 Introduction .......................................... 6 1.1 Overview ........................................ 6 1.2 Grouptheory ..................................... 8 1.3 Graphtheory ..................................... 16 1.4 Coxetergroups ................................... 27 2 Local recognition of graphs .............................. 41 2.1 Somelocalrecognitionresults . ....... 43 2.2 Local recognition of simply laced Weyl graphs . ........ 45 2.3 Local recognition of non-simply laced Weyl graphs . ......... 48 2.4 Graphs locally like W( F4) .............................. 50 2.5 Recognition results for graphs locally like W( F4) ................ 53 3 Group theoretic applications ............................. 65 3.1 Recognizing Symn ................................... 66 3.2 Recognizing Coxeter groups . ..... 68 3.3 Outlook: Applications to Chevalley groups . ......... 75 Appendix A Computer code ............................... 80 A.1 ComputationsinSAGE .............................. 80 A.2 ComputationsinGAP ............................... 88 References ............................................ 89 Index ...............................................
    [Show full text]
  • Efficient Algorithms for Graphs with Few P-4'S
    Old Dominion University ODU Digital Commons Computer Science Faculty Publications Computer Science 2001 Efficient Algorithms for Graphs with Few P-4’s Luitpold Babel Ton Kloks Jan Kratochvíl Dieter Kratsch Kaiko Müller See next page for additional authors Follow this and additional works at: https://digitalcommons.odu.edu/computerscience_fac_pubs Part of the Computer Sciences Commons, and the Mathematics Commons Repository Citation Babel, Luitpold; Kloks, Ton; Kratochvíl, Jan; Kratsch, Dieter; Müller, Kaiko; and Olariu, Stephan, "Efficient Algorithms for Graphs with Few P-4’s" (2001). Computer Science Faculty Publications. 104. https://digitalcommons.odu.edu/computerscience_fac_pubs/104 Original Publication Citation Babel, L., Kloks, T., Kratochvil, J., Kratsch, D., Muller, H., & Olariu, S. (2001). Efficient algorithms for graphs with few P4's. Discrete Mathematics, 235(1-3), 29-51. doi:10.1016/s0012-365x(00)00258-2 This Article is brought to you for free and open access by the Computer Science at ODU Digital Commons. It has been accepted for inclusion in Computer Science Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Authors Luitpold Babel, Ton Kloks, Jan Kratochvíl, Dieter Kratsch, Kaiko Müller, and Stephan Olariu This article is available at ODU Digital Commons: https://digitalcommons.odu.edu/computerscience_fac_pubs/104 Discrete Mathematics 235 (2001) 29–51 www.elsevier.com/locate/disc Ecient algorithms for graphs with few P4’s Luitpold Babela;∗, Ton Kloksb;1, Jan KratochvÃ.lb;2, Dieter Kratschc, Haiko Muller0 c, Stephan Olariud aZentrum Mathematik, Technische Universitat Munchen, 80290 Munchen, Germany bDIMATIA, Charles University, 118 00 Praha 1, Czech Republic cFriedrich-Schiller-Universitat Jena, Fakultat fur Mathematik und Informatik, 07740 Jena, Germany dDepartment of Computer Science, Old Dominion University, Norfolk, VA 23529, USA Abstract We show that a large variety ofNP-complete problems can be solved eciently forgraphs with ‘few’ P4’s.
    [Show full text]
  • Operator Decomposition of Graphs
    152 The International Arab Journal of Information Technology, Vol. 3, No. 2, April 2006 Operator Decomposition of Graphs Ruzayn Quaddoura Computer Science Department, Zarqa Private University, Jordan Abstract: In this paper we introduce a new form of decomposition of graphs, the (P, Q) -decomposition. We first give an optimal algorithm for finding the 1 -decomposition of a graph which is a special case of the (P, Q) -decomposition which was first introduced in [ 21 ]. We then examine the connections between the 1 -decomposition and well known forms of decomposition of graphs, namely, modular and homogeneous decomposition. The characterization of graphs totally decomposable by 1 -decomposition is also given. The last part of our paper is devoted to a generalization of the 1 - decomposition. We first show that some basic properties of modular decomposi tion can be extended in a new form of decomposition of graphs that we called operator decomposition. We introduce the notion of a (P, Q) -module, where P and Q are hereditary graph -theoretic properties, the notion of a (P, Q) -split graph and the closed here ditary class (P, Q) of graphs (P and Q are closed under the operations of join of graphs and disjoint union of graphs, respectively). On this base , we construct a special case of the operator decomposition that is called (P, Q) -decomposition. Such decompos ition is uniquely determined by an arbitrary minimal nontrivial (P, Q) -module in G. In particular, if G ( P, Q) , then G has the unique canonical (P, Q) -decomposition. Keywords: Graph decomposition, hereditary class, split graph .
    [Show full text]
  • 1.2 Graph Laplacians
    -Mm Active Flows and Networks by Aden Forrow Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 0 Massachusetts Institute of Technology 2018. All rights reserved. Signature redacted A u th or ................................ Department of Mathematics May 4, 2018 Signature redacted C ertified by ............................. J6rn Dunkel Assistant Professor Thesis Supervisor Accepted by ................. Signature redacted Jonathan Kelner Chairman, Department Committee on Graduate Theses MASSACHUSES INSTITUTE OF TECHNOLOGY MAY 3 0 2018 LIBRARIES ARCHIVES 2 Active Flows and Networks by Aden Forrow Submitted to the Department of Mathematics on May 4, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Coherent, large scale dynamics in many nonequilibrium physical, biological, or in- formation transport networks are driven by small-scale local energy input. In the first part of this thesis, we introduce and explore two analytically tractable nonlinear models for such active flow networks, drawing motivation from recent microfluidic experiments on bacterial and other microbial suspensions. In contrast to equiparti- tion with thermal driving, we find that active friction selects discrete states with only a limited number of modes excited at distinct fixed amplitudes. When the active transport network is incompressible, these modes are cycles with constant flow; when it is compressible, they are oscillatory. As is common in such network dynamical sys- tems, the spectrum of the underlying graph Laplacian plays a key role in controlling the flow. Spectral graph theory has traditionally prioritized analyzing Laplacians of unweighted networks with specified adjacency properties.
    [Show full text]
  • Shrub-Depth and M-Partite Cographs?
    When Trees Grow Low: Shrub-depth and m-Partite Cographs? Robert Ganian1, Petr Hlinˇen´y2, Jaroslav Neˇsetˇril3, Jan Obdrˇz´alek2, and Patrice Ossona de Mendez4 1 Vienna University of Technology, Austria [email protected] 2 Faculty of Informatics, Masaryk University, Brno, Czech Republicy fhlineny,[email protected] 3 Computer Science Inst. of Charles University (IUUK), Praha, Czech Republicy;z [email protected] 4 CNRS UMR 8557, Ecole´ des Hautes Etudes´ en Sciences Sociales, Paris, Francez [email protected] Abstract. The recent increase of interest in the graph invariant called tree-depth and in its algorithmic applications leads to a natural question: Is there an analogously useful \depth" notion also for dense graphs (say; one which is stable by graph complementation)? To this end we intro- duce the notion of shrub-depth of a graph class which is related to the established notion of clique-width in a similar way as tree-depth is re- lated to tree-width. We also introduce a common extension of cographs and of classes with bounded shrub-depth | m-partite cographs, which are well-quasi-ordered by the relation \being an induced subgraph of" and therefore allow polynomial time testing of any hereditary properties. 1 Introduction In this paper, we are interested in structural graph parameters that are interme- diate between clique-width and tree-depth, sharing the nice properties of both. Clique-width, defined in [7], is the older of the two notions. In several aspects, the theory of graphs of bounded clique-width is similar to the one of bounded tree- width.
    [Show full text]
  • Defining Recursive Predicates in Graph Orders
    Logical Methods in Computer Science Vol. 14(3:21)2018, pp. 1–38 Submitted Sep. 12, 2017 https://lmcs.episciences.org/ Published Sep. 24, 2018 DEFINING RECURSIVE PREDICATES IN GRAPH ORDERS RAMANATHAN S. THINNIYAM The Institute of Mathematical Sciences, 4th Cross Street, CIT Campus, Tharamani, Chennai, Tamil Nadu 600113, India Homi Bhabha National Institute, Mumbai, India e-mail address: [email protected] Abstract. We study the first order theory of structures over graphs i.e. structures of the form (G; τ) where G is the set of all (isomorphism types of) finite undirected graphs and τ some vocabulary. We define the notion of a recursive predicate over graphs using Turing Machine recognizable string encodings of graphs. We also define the notion of an arithmetical relation over graphs(borrowed from [20]) using a total order ≤t on the set G such that (G; ≤t) is isomorphic to (N; ≤). We introduce the notion of a capable structure over graphs, which is an arithmetical structure satisfying the conditions : (1) definability of arithmetic, (2) definability of cardinality of a graph, and (3) definability of two particular graph predicates related to vertex labellings of graphs. We then show any capable structure can define every arithmetical predicate over graphs. As a corollary, any capable structure also defines every recursive graph relation. We identify capable structures which are expansions of graph orders, which are structures of the form (G; ≤) where ≤ is interpreted as a partial order. We show that the subgraph order i.e. (G; ≤s), induced subgraph order with one constant P3 i.e. (G; ≤i;P3) and an expansion of the minor order for counting edges i.e.
    [Show full text]