Introduction To Graph Convexity
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Introduction to Graph Convexity
This book focuses on the computational aspects of graph convexity, with a particular emphasis on path convexity within graphs. It provides a thoughtful introduction to this emerging research field, which originated by adapting concepts from convex geometry to combinatorics and has experienced substantial growth. The book starts with an introduction of fundamental convexity concepts and then proceeds to discuss convexity parameters. These parameters fall into two categories: one derived from abstract convexity studies and another motivated by computational complexity. Subsequent chapters explore geometric convexity within graphs, examining various graph classes such as interval graphs, proper interval graphs, cographs, chordal graphs, and strongly chordal graphs. The text concludes with a study of the computation of convexity parameters across different convexity types, including practical applications in areas like game theory. Compact and straightforward, this work serves as an ideal entry point for students and researchers interested in pursuing further research in the field of convexity. The English translation of this book, originally in Portuguese, was facilitated by artificial intelligence. The content was later revised by the authors for accuracy.
Convex Optimization
Author: Stephen P. Boyd
language: en
Publisher: Cambridge University Press
Release Date: 2004-03-08
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
Convexity in Graphs
A natural concept of convexity for directed graphs is introduced, and properties of the lattice of convex subgraphs of a graph are studied. The extent to which this lattice determines the graph is established, and conditions for a lattice to be a convex subgraph lattice are investigated. The concept of a lower semi-homomorphism is defined for lattices; it is shown that such mappings preserve basic properties of convex subgraph lattices, and that on such lattices, they are uniquely determined by their kernels. Graph homomorphisms which preserve convexity are also studied, with emphasis on their relationship to lower semi-homomorphisms of the convex subgraph lattice. Homomorphisms which 'contract' subgraphs (which are analogous to the rewriting rules of context-sensitive phrase structure grammars) are briefly considered. Finally, a concept of local convexity for directed graphs is introduced. (Author).