Graph theory is a delightful playground for the exploration ofproof tech-niques in discrete mathematics, and its results haveapplications in many areas of the computing, social, and naturalsciences. The design of this book permits usage in a one-semesterintroduction at the undergraduate or beginning grad-uate level, orin a patient two-semester introduction. No previous knowledge ofgraph theory is assumed. Many algorithms and applicaions areincluded, but the focus is on understanding the structure of graphsand the techniques used to analyze problems in graph theory.
Many textbooks have been written about graph theory. Due to itsem-phasis on both proofs and applications, the initial model forthis book was the elegant text by J.A. Bondy and U.S.R. Murty,Graph Theory with Applica-tions (Macmillan/North-Holland [1976]).Graph theory is still young, and no consensus has emerged on howthe introductory material should be presented. Selection and orderof topics, choice of proofs, objectives, and underlying themes arematters of lively debate. Revising this book dozens of times hastaught me the difficulty of these decisions. This book is mycontribution to the debate.
Introduction To Graph Theory Douglas West Pdf
Download File: https://3menpi0ili.blogspot.com/?download=2vHT7s
Many undergraduates begin a course in graph theory with littleexposure to proof techniques. Appendix A provides backgroundreading that will help them get started. Students who havedifficulty understanding or writing proofs in the early materialshould be encouraged to read this appendix in conjunction withChapter 1. Some discussion of proof techniques still appears in theearly sections of the text (especially concerning induction), butan expanded treat-ment of the basic background (especiallyconcerning sets, functions, relations, and elementary counting) isnow in Appendix A.
Most of the exercises require proofs. Many undergraduates havehad lit-tle practice at presenting explanations, and this hinderstheir appreciation of graph theory and other mathematics. Theintellectual discipline of justifying an argument isvaluable.independently of mathematics; I hope that students willappreciate this. In writing solutions to exercises, students shouldbe careful in their use of language ("say what you mean"), and theyshould be intellectually honest ("mean what you say").
Although many terms in graph theory suggest their definitions,the quan-tity of terminology remains an obstacle to fluency.Mathematicians like to gather definitfons at the start, but moststudents succeed better if they use a
I have changed the treatment of digraphs substantially bypostponing their introduction to Section 1.4. Introducing digraphsat the same time as graphs tends to confuse or overwhelm students.Waiting to the end of Chapter 1 al-lows them to become comfortablewith basic concepts in the context of a single model. Thediscussion of digraphs then reinforces some of those concepts whileclarifying the distinctions. The two models are still discussedtogether in the material on connectivity.
This book contains more material than most introductory texts ingraph theory. Collecting the advanced material as a final optionalchapter of "addi-tional topics" permits usage at different levels.The undergraduate introduction consists of the first seven chapters(omitting most optional material), leaving Chapter 8 as topicalreading for interested students. A graduate course can treat mostof Chapters 1 and 2 as recommended reading, moving rapidly toChapter 3 in class and reaching some topics in Chapter 8. Chapter 8can also be used as the basis for a second course in graph theory,along with material that was optional in earlier chapters.
Many results in graph theory have several proofs; illustratingthis can in-crease students' flexibility in trying multipleapproaches to a problem. I include some alternative proofs asremarks and others as exercises.
In addition to the material on isomorphism, Section 1.1 now hasa more precise treatment of the Petersen graph and an explicitintroduction of the notions of decomposition and girth. Thisprovides language that facilitates later discussion in variousplaces, and it permits interesting explicit questions other thanisomorphism.
Sections 1.2-1.4 have become more coherent. The treatment ofEulerian circuits motivates and completes Section 1.2. Somematerial has been removed from Section 1.3 to narrow its focus todegrees and counting, and this section has acquired the material onvertex degrees that had been in Section 1.4. Sec-tion 1.4 nowprovides the introduction to digraphs and can be treatedlightly.
The text explores relationships among fundamental results.Petersen's Theorem on 2-factors (Chapter 3) uses Eulerian circuitsand bipartite match-ing. The equivalence between Menger's Theoremand the Max Flow-Min Cut Theorem is explored more fully than in thefirst edition, and the "Baseball Elim-ination" application is nowtreated in more depth. The k - !-connectedness of k-color-criticalgraphs (Chapter 5) uses bipartite matching. Section 5.3 offers abrief introduction to perfect graphs, emphasizing chordal graphs.Additional features of this text in comparison to some othersinclude the algorithmic proof of Vizing's Theorem and the proof ofKuratowski's Theorem by Thomassen's methods.
I will treat advanced graph theory more thoroughly in The Art ofCombina-torics. Volume I is devoted to extremal graph theory andVolume II to structure of graphs. Volume III has chapters onmatroids and integer programming (in-cluding network flow). VolumeIV emphasizes methods in cpmbinatorics and discusses variousaspects of graphs, especially random graphs.
In chapters after the first, the most fundamental material isconcentrated in the first section. Emphasizing these sections(while skipping the optional items) still illustrates the scope ofgraph theory in a slower-paced one-semester course. From the secondsections of Chapters 2, 4, 5, 6, and 7, it would be bene-ficial toinclude Cayley's Formula, Menger's Theorem, Mycielski'sconstruction, Kuratowski's Theorem, and Dirac's Theorem (spanningcycles), respectively.
Some optional material is particularly appealing to present inclass. For example, I always present the optional subsections onDisjoint Spanning Trees (in Section 2.1) and Stable Matchings (inSection 3.2), and I usually present the optional subsection on/-factors (in Section 3.3). Subsections are marked optional when nolater material in the first seven chapters requires them and theyare not part of the central development of elementary graph theory,but these are nice applications that engage students' interest. Inone sense, the "optional" marking indicates to students that thefinal examination is unlikely to have questions on thesetopics.
Courses that introduce graph theory in one term under thequarter system must aim for highlights; I suggest the followingrough syllabus: 1.1: adjacency matrix, isomorphism, Petersen graph.1.2: all. 1.3: degree-sum formula, large bipartite S,ubgraphs. 1.4:up to strong components, plus tournaments. 2.1: up
Another theme that underlies much of Chapters 3-5 and Section 7.1 is that of dual maximization and minimization problems. In agraph theory course one does not want to delve deeply into thenature of duality in linear optimization. It suffices to say thattwo optimization problems form a dual pair when every feasiblesolution to the maximization problem has value at most the value ofevery feasible solution to the minimization problem. When feasiblesolutions with the same value are given for the two problems, thisduality implies that both solutions are optimal. A discussion ofthe linear programming context appears in Section 8.1.
Due to its reduced emphasis on numerical computation andincreased em-phasize on techniques of proof and clarity ofexplanations, graph theory is an excellent subject in which toencourage students to improve their habits of communication, bothwritten and oral. In addition to assigning written home-work thatrequires carefully presented arguments, I have found it productiveto organize optional "collaborative study" sessions in whichstudents can work together on problems while I circulate, listen,and answer questions. It should not be forgotten that one of thebest ways to discover whether one understands a proof is to try toexplain it to someone else. The students who participate find thesesessions very beneficial.
west/igt I have corrected alltypographical and mathematical errors known to me be-fore the timeof printing. Nevertheless, the robustness of the set of errors andthe substantial rewriting and additions make me confident thRt someerror remains. Please find it and tell me so I can correct it!
These and many other practical problems involve graph theory. Inthis book, we develop the theory of graphs and apply it to suchproblems. Our starting point assumes the mathematical background inAppendix A, where basic objects and language of mathematics arediscussed.
Before embarking on this, we review an important technique ofproof Many statements in graph theory can be proved using theprinciple of induction. Readers unfamiliar with induction shouldread the material on this proof tech-nique in Appendix A. Here wedescribe the form of induction that we will use most frequently, inorder to familiarize the reader with a template for proof.
I am interested in learning graph theory, and from many resources I came to know that Douglas West's Introduction to Graph Theory is a good textbook. But since I am doing self-study, it is at times hard to understand the material without proper guidance.
So if anyone knows about any video lecture series on graph theory which uses the said text for reference, then it will be quite helpful to me. It would be like taking an actual course, getting input from the instructor, and then improvising on the learning thereafter by reading the text.
Course DescriptionThis course will cover the fundamental concepts of graph theory: simple graphs, digraphs, Eulerian and Hamiltonian graphs, trees, matchings, networks, paths and cycles, graph colorings, and planar graphs. Famous problems in graph theory include: the Minimum Connector Problem (building roads at minimum cost), the Marriage Problem (matching men and women into compatible pairs), the Assignment Problem (filling jobs with applicants), the Network Flow Problem (maximizing flow in a network), the Committee Scheduling Problem (using the fewest time slots), the Four Color Problem (coloring maps with four colors so that adjacent regions have different colors), and the Traveling Salesman Problem (visiting a list of cities minimizing the traveled distance). 2ff7e9595c
Comments