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Intersection Graphs of Boxes and Cubes

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Intersection Graphs of Boxes and Cubes Intersection graphs of boxes and cubes A Thesis Submitted For the Degree of Doctor of Philosophy in the Faculty of Engineering by Mathew C. Francis Department of Computer Science and Automation Indian Institute of Science Bangalore – 560 012 July, 2009 To my parents and all my teachers Acknowledgements Of all people, I should thank Dr. L. Sunil Chandran first, as the work behind this thesis is as much his as it is mine. The faith he reposed in me was at times as puzzling to me as it was reassuring. I am indebted to Dr. Naveen Sivadasan for the long discussions we had that not only produced results but went a long way in helping me learn the ropes. The brief but fruitful collaboration with Santhosh Suresh was thoroughly enjoyable. I am thankful to Dr. Samir Datta for his insights on planar graphs. The stimulating discussions with Dr. Irith Hartman, Rogers, Manu, Abhijin, Anita, Meghna, Sadagopan, Chintan and Subramanya have helped shape my view of the subject. Words cannot express my gratitude towards all my friends at IISc, each one of them inimitable, each one with a different perspective of the world but at the same time car- ing, guiding and helping with all their hearts. Rogers, Raj Mohan, Murali Sir, Sheron, Thomas, Ashik, Dileep, Shijo, Hari, Deepak Ravi, Rashid have all left indelible impres- sions on me. I am grateful to Nicky for her care and understanding. It is impossible to thank my parents enough for their unflinching support and constant encouragement. i Abstract A graph G is said to be an intersection graph of sets from a family of sets if there exists F a function f : V (G) such that for u, v V (G), (u, v) E(G) f(u) f(v) = → F ∈ ∈ ⇔ ∩ 6 . Interval graphs are thus the intersection graphs of closed intervals on the real line ∅ and unit interval graphs are the intersection graphs of unit length intervals on the real line. An interval on the real line can be generalized to a “k-box” in Rk.A k-box B = (R1,R2,...,Rk), where each Ri is a closed interval on the real line, is defined to be the Cartesian product R R R . If each R is a unit length interval, we 1 × 2 ×···× k i call B a k-cube. Thus, 1-boxes are just closed intervals on the real line whereas 2-boxes are axis-parallel rectangles in the plane. We study the intersection graphs of k-boxes and k-cubes. The parameter boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. Thus, interval graphs are the graphs with boxicity at most 1 and unit interval graphs are the graphs with cubicity at most 1. These parameters were introduced by F. S. Roberts in 1969. In some sense, the boxicity of a graph is a measure of how different a graph is from an interval graph and in a similar way, the cubicity is a measure of how different the graph is from a unit interval graph. We prove several upper bounds on the boxicity and cubicity of general as well as special classes of graphs in terms of various graph parameters such as the maximum degree, the number of vertices and the bandwidth. The following are some of the main results presented. 1. We show that for any graph G with maximum degree ∆, box(G) 2∆2. This ≤ ii iii result implies that bounded degree graphs have bounded boxicity no matter how large the graph might be. 2. It was shown in [18] that the boxicity of a graph on n vertices with maximum degree ∆ is O(∆ln n). But a similar bound does not hold for the average degree dav of a graph. [18] gives graphs in which the boxicity is exponentially larger than dav ln n. We show that even though an O(dav ln n) upper bound for boxicity does not hold for all graphs, for almost all graphs, boxicity is O(dav ln n). 3. The ratio of the cubicity to boxicity of any graph shown in [15] when combined with the results on boxicity show that cub(G) is O(∆ln2 n) and O(∆2 ln n) for any graph G on n vertices and with maximum degree ∆. By using a randomized construction, we prove the better upper bound cub(G) 4(∆ + 1) ln n . ≤ ⌈ ⌉ 4. Two results relating the cubicity of a graph to its bandwidth b are presented. First, it is shown that cub(G) 12(∆ + 1) ln(2b) +1. Next, we derive the upper bound ≤ ⌈ ⌉ cub(G) b + 1. This bound is used to derive new upper bounds on the cubicity of ≤ special graph classes like circular arc graphs, cocomparability graphs and AT-free graphs in relation to the maximum degree. 5. The upper bound for cubicity in terms of the bandwidth gives an upper bound of ∆ + 1 for the cubicity of interval graphs. This bound is improved to show that for any interval graph G with maximum degree ∆, cub(G) log ∆ + 4. ≤ ⌈ 2 ⌉ 6. Scheinerman [54] proved that the boxicity of any outerplanar graph is at most 2. We present an independent proof for the same theorem. 7. Halin graphs are planar graphs formed by adding a cycle connecting the leaves of a tree none of whose vertices have degree 2. We prove that the boxicity of any Halin graph is equal to 2 unless it is a complete graph on 4 vertices, in which case its boxicity is 1. Publications based on this thesis 1. “Geometric representation of graphs in low dimension using axis-parallel boxes”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, accepted for publi- cation in Algorithmica, doi:10.1007/s00453-008-9163-5, 2008. 2. “Boxicity and maximum degree”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, Journal of Combinatorial Theory, Series B, 98(2):443–445, March 2008. 3. “Representing graphs as the intersection of axis-parallel cubes”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, MCDES 2008, Bangalore, May 2008. 4. “On the cubicity of AT-free graphs and circular-arc graphs”, L. Sunil Chandran, Mathew C. Francis and Naveen Sivadasan, Graph Theory, Computational Intelli- gence and Thought, Israel, September 2008. 5. “On the cubicity of interval graphs”, Graphs and Combinatorics, 25(2):169–179, May 2009. 6. “Boxicity of Halin graphs”, Discrete Mathematics, 309(10):3233–3237, May 2009. iv Contents Acknowledgements i Abstract ii Publications based on this thesis iv 1 Introduction 1 1.1 Basicdefinitionsandnotations. 1 1.2 Intervalgraphsandboxicity . 3 1.2.1 k-boxes: intervals in higher dimensions . 5 1.2.2 Boxicity ................................ 7 1.2.3 Interval graph representation of a graph . 8 1.3 Unitintervalgraphsandcubicity . 10 1.3.1 Unit and equal interval representations as mappings to real numbers 11 1.3.2 k-cubes ................................ 12 1.3.3 Cubicity................................ 13 1.3.4 Indifference graph representation of a graph . 13 1.4 Anoteontheasymptoticnotation . 14 1.5 Ashortsurveyofpreviousliterature . 14 1.5.1 Resultsonboxicity .......................... 15 1.5.2 Boxicity in other scientific disciplines . 16 1.5.3 Resultsoncubicity .......................... 17 1.5.4 Other geometric intersection graph classes . 18 1.6 Outlineoftherestofthethesis . 18 2 Upper bounds for boxicity 21 2.1 Previousupperboundsonboxicity . 21 2.1.1 Boxicity is O(∆ln n)......................... 21 2.1.2 Boxicity and average degree . 22 2.2 Boxicity of bounded degree graphs . 22 2.3 Concludingremarks.............................. 24 v CONTENTS vi 3 Boxicity of random graphs 27 3.1 Randomgraphpreliminaries . 27 3.2 Boxicity is O(dav ln n)foralmostallgraphs. 28 3.3 Remarks.................................... 31 4 A randomized construction for cubicity 33 4.1 The algorithm RAND ............................ 34 4.2 Derandomizing RAND ........................... 39 4.3 Ausefulresult................................. 47 5 Cubicity and bandwidth 49 5.1 Cube representation in O(∆ln b)dimensions ................ 50 5.2 Cube representation in b +1dimensions .................. 55 5.3 Cubicity of special graph classes . 59 5.3.1 Circular-arcgraphs . 59 5.3.2 Cocomparabilitygraphs . 61 5.3.3 AT-freegraphs ............................ 62 5.4 Asummaryofresults............................. 63 6 Cubicity of interval graphs 65 6.1 Afewresultsthatweneed.......................... 65 6.2 Theproof ................................... 67 6.3 Remarks.................................... 74 7 Planar graphs 77 7.1 Preliminaries ................................. 77 7.2 Outerplanargraphs.............................. 79 7.3 Discussion................................... 80 8 Boxicity of Halin graphs 81 8.1 Ashortintroduction ............................. 81 8.2 Theproof ................................... 82 8.3 Results..................................... 90 9 Conclusion 91 9.1 Improvements................................. 91 9.2 Openproblems ................................ 92 9.3 Endnote .................................... 93 Bibliography 96 List of Figures 1.1 Anexampleofanintervalgraph. 3 1.2 Anasteroidaltriple.............................. 4 1.3 A 2-box in R2 and a 3-box in R3 ....................... 6 1.4 A 2-box representation for C4 ........................ 7 1.5 K1,n, the star graph with n arms ...................... 10 2.1 Structure of Gi ................................ 23 5.1 Acircular-arcgraph ............................. 59 5.2 Anexampleofacaterpillar ......................... 62 7.1 A book drawing of K5 using3pages .................... 78 8.1 AHalingraph................................. 81 vii Chapter 1 Introduction All graphs considered in this work will be simple, undirected and finite. Most of the graph theoretic notations used shall be defined in the following section. Much of it has been borrowed from the book “Graph Theory” by Reinhard Diestel [26]. The reader may please refer to Chapter 1 of [26] for any notations that are not defined here. 1.1 Basic definitions and notations The notations G(V,E), G =(V,E) or simply G will be used to indicate a graph G which has a vertex set V (G) and an edge set E(G).
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