LINE GRAPHS AND FLOW NETS A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Bridget Kinsella Druken Fall 2010 iii Copyright c 2010 by Bridget Kinsella Druken iv The man in the wilderness asked of me, “How many strawberries grow in the salt sea?” I answered him, as I thought good, As many a ship as sails in the wood. The man in the wilderness asked me why His hen could swim and his pig could fly. I answered him as I thought best, “They were both born in a cuckoos nest.” The man in the wilderness asked me to tell All the sands in the sea and I counted them well. He said he with a grin, “And not one more?” I answered him, “Now you go make sure.” – written by Mother Goose, sung by Natalie Merchant v ABSTRACT OF THE THESIS Line Graphs and Flow Nets by Bridget Kinsella Druken Master of Arts in Mathematics San Diego State University, 2010 Every graph has a line graph, but not all graphs are line graphs of some graph. Certain types of induced subgraphs indicate whether a graph is a line graph. For example, the claw is not a line graph. This thesis gives a careful characterization of the main theorem of line graphs, that is, the necessary and sufficient conditions for a graph to be a line graph. We show the relationship between triangles, forbidden subgraphs, Krausz partitions and line graphs by providing additional lemmas which aim to support the main theorem. Less attention has been devoted to line graphs of directed graphs. De Bruijn graphs give a motivation to investigate this class since they are directed line graphs. Another motivation for learning about directed line graphs arises from the study of the sum-product algorithm for low density parity check codes. We work with a very general class of directed graphs that allow multiple edges and loops, which are called nets by Harary. We investigate the flow net and compare it to the line graph to lay the foundation for future study of directed line graphs. vi TABLE OF CONTENTS PAGE ABSTRACT .................................................................................... v LIST OF TABLES.............................................................................. vii LIST OF FIGURES ............................................................................ viii ACKNOWLEDGEMENTS.................................................................... x CHAPTER 1 GRAPHSANDNETS................................................................. 1 1.1 Graphs ........................................................................... 2 1.2 Nets .............................................................................. 5 1.3 Matrix Representation .......................................................... 7 2 LINEGRAPHS ........................................................................ 10 2.1 Triangles......................................................................... 15 2.2 Forbidden Subgraphs............................................................ 18 2.3 Krausz Partitions and Reconstructing the Root Graph ........................ 22 2.4 MainTheorem for LineGraphs ................................................ 28 3 FLOWNETS........................................................................... 30 3.1 DeBruijnGraphs................................................................ 30 3.2 LineNetsandFlowNets........................................................ 34 3.3 FlowNetsandLineNets........................................................ 38 4 CONCLUSION ........................................................................ 39 BIBLIOGRAPHY.............................................................................. 40 APPENDIX KRAUSZ RECOVERING PROCESS APPLIED TO THE NINE FORBID- DEN SUBGRAPHS ................................................................... 41 vii LIST OF TABLES PAGE Table A.1 Applying Krausz Recovering Process to the Nine Forbidden Subgraphs. ....... 42 viii LIST OF FIGURES PAGE Figure 1.1 Induced subgraphs and spanning subgraphs....................................... 4 Figure 1.2 A graph G and a net N............................................................. 7 Figure 2.1 Graphs and their line graphs. ...................................................... 11 Figure 2.2 All connected graphs with three edges and their linegraphs..................... 12 Figure 2.3 All connected graphs with four edges and their linegraphs...................... 12 Figure 2.4 Example of a triangle t1t2t3 made odd by vertex u in K5. ....................... 16 Figure 2.5 Nine forbidden subgraphs first found by Beineke in 1968........................ 18 Figure 2.6 Three forbidden subgraphs which result from Case 1. ........................... 19 Figure 2.7 Two forbidden subgraphs which result from Case 2a............................. 20 Figure 2.8 Two forbidden subgraphs which result from Case 2b............................. 20 Figure 2.9 Two forbidden subgraphs which result from Case 2c. ........................... 21 Figure 2.10 Four forbidden subgraphs which result from Case 2d........................... 21 Figure 2.11 Two forbidden subgraphs which result from Case 2e. .......................... 21 Figure 2.12 Two forbidden subgraphs which result from Case 2f............................ 22 Figure 2.13 (a) Line graph L with a Krausz partition, (b) root graph G. .................... 23 Figure 2.14 Applying the Krausz recovering process to line graph L........................ 25 Figure 2.15 All ways two triangles that share an edge can be even. ......................... 25 Figure 2.16 Case 3: x ∈ D1 ∩ D2 ∩ C1....................................................... 27 Figure 3.1 Examples of De Bruijn sequences of order 2, 3 and 4 with symbols 0, 1........ 31 Figure 3.2 De Bruijn graph(2, 3) with Hamiltonian circuit................................... 32 Figure 3.3 De Bruijn graph(2, 2) with Eulerian circuit. ...................................... 32 Figure 3.4 De Bruijn graph(2, 4) with Hamiltonian circuit................................... 33 Figure 3.5 Examples of a symmetric net N of K1,3 and its line net L(N)................... 35 Figure 3.6 Examples of a symmetric net N of C3 and its line net L(N). ................... 35 ∗ Figure 3.7 Examples of a symmetric net N of C3 and its flow net N ....................... 36 ∗ Figure 3.8 Examples of a symmetric net N of K1,3 and its flow net N ..................... 36 Figure 3.9 Example of a connected net with a disconnected linenet. ....................... 37 Figure 3.10 Example of two nets where N 6∼= N ′ but L(N) ∼= L(N ′)....................... 37 ix Figure 3.11 Example of a net N ∼= L(N)..................................................... 37 Figure A.1 G1, K(G1), L(K(G1))............................................................. 42 Figure A.2 G2, K(G2), L(K(G2))............................................................. 43 Figure A.3 G3, K(G3), L(K(G3))............................................................. 43 Figure A.4 G4, K(G4), L(K(G4))............................................................. 43 Figure A.5 G5, K(G5), L(K(G5))............................................................. 43 Figure A.6 G6, K(G6), L(K(G6))............................................................. 44 Figure A.7 G7, K(G7), L(K(G7))............................................................. 44 Figure A.8 G8, K(G8), L(K(G8))............................................................. 44 Figure A.9 G9, K(G9), L(K(G9))............................................................. 44 x ACKNOWLEDGEMENTS I would like to thank Dr. Michael O’Sullivan for his support, feedback and for being a wonderful mentor throughout the course of my thesis work. I would like to thank Dr. Roxana Smarandache and Dr. Joseph Lewis for serving on my committee and providing insightful feedback. I would also like to thank Dr. Andrew Izs´ak, Dr. Joanne Lobato, and Dr. Susan Nickerson for providing teaching support and research opportunities in math education. A big thanks to my dear friends, fellow graduate students, nice professors and helpful staff who made my graduate school experience enjoyable and rich. My learning has been a product of my interactive environment and self-reflection. I would like to thank my awesome family Val, Pat, Kelsey, Abbey, Catherine, Glenna and Maddie, whether near or far away in Rhode Island, for their support, laughter, stories, encouragement and perspective. I am lucky to have such loving, kind and thoughtful sisters. In particular, I would like to thank: my sister Kelsey Druken for her wise words regarding academia, support and being an amazing role model; Mell Mol for her perspective, contagious happiness, surf sessions, and sharing both her dog Zona and car with me; and Natalie Selinski for her artistic and flexible interpretations of mathematics, support for teaching and mathematics education, flaps and being a great human. I would like to thank xkcd.com, PhdComics.com, Huffington Post and NPR for providing much welcomed entertainment, distraction and information. 1 CHAPTER 1 GRAPHS AND NETS One definition of a graph is a finite nonempty set V of vertices with a set E of edges or unordered pairs of vertices, without multiple edges, loops or directed edges [1]. To make this point clear, authors will call it a “simple graph.” A graph that allows multiple edges is called a multigraph. A graph that allows multiple edges and loops is called a pseudograph, although some authors use multigraph and pseudograph interchangeably.