Without knowledge of the benefits of (automatic) graph drawing, be sure to not miss out on "Preface."
The part on yFiles Basic is comprised of the graph analysis algorithms chapter and the description of the basic algorithms graph structure, which is in Chapter 6, Underlying Graph Model of the yFiles Graph Analysis Algorithms and Chapter 7, Working with the Algorithms Graph Structure. These latter two chapters are necessary for more involved graph analysis tasks, but can be easily skipped, initially.
Chapter 4, Graph Analysis that presents graph analysis algorithms, gives a quick introduction how to run them in conjunction with an IGraph-based graph in the section called “Using yFiles Graph Analysis Functionality”. This section should be read before any section on actual graph analysis algorithms.
The part on yFiles Layout ("Automatic Graph Layout") explains the layout concepts and provides in-depth information concerning all layout algorithms. At the beginning of this chapter, "Using yFiles Layout Functionality" gives a quick introduction how to conveniently run the layout algorithms in conjunction with an IGraph-based graph. It should be read before any sections on actual layout algorithms.
"Using yFiles for Silverlight Algorithms Functionality" discusses the adapter mechanisms for calling layout algorithms and graph analysis algorithms in conjunction with an IGraph-based graph in detail. Initially, this chapter can be safely skipped.
The chapter "Underlying Graph Model of the yFiles Layout Algorithms" presents the LayoutGraph API, which provides the means for advanced customizations of a layout process that go beyond the support of the adapter mechanisms. As such, the chapter is needed in specific situations only.
Lastly, the glossary cannot be consulted often enough. It explains common graph structure related terminology as well as all terms specifically used with yFiles.
Appendix A, Obfuscation is a must-read concerning the actual deployment of any product developed using the yFiles for Silverlight library. It discusses the code protection requirements of the yFiles for Silverlight license terms and helps in understanding how to fulfill them properly.
The information in Appendix B, License Mechanism is equally important, but relevant also for software development since it explains how to use yFiles for Silverlight license files in yFiles for Silverlight-based applications.
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