图论和超图理论简介

English | 2013 | ISBN: 1606923722 | PDF | pages: 303 | 9.2 mb

This book is for math and computer science majors, for students and representatives of many other disciplines (like bioinformatics, for example) taking courses in graph theory, discrete mathematics, data structures, algorithms. It is also for anyone who wants to understand the basics of graph theory, or just is curious. No previous knowledge in graph theory or any other significant mathematics is required. The very basic facts from set theory, proof techniques and algorithms are sufficient to understand it; but even those are explained in the text. Structurally, the text is divided into two parts where Part II is the generalization of Part I.The first part discusses the key concepts of graph theory with emphasis on trees, bipartite graphs, cycles, chordal graphs, planar graphs and graph coloring. The second part considers generalizations of Part I and discusses hypertrees, bipartite hyper graphs, hyper cycles, chordal hyper graphs, planar hyper graphs and hyper graph coloring. There is an interaction between the parts and within the parts to show how ideas of generalizations work. The main point is to exhibit the ways of generalizations and interactions of mathematical concepts from the very simple to the most advanced. One of the features of this text is the duality of hyper graphs.This fundamental concept is missing in graph theory (and in its introductory teaching) because dual graphs are not properly graphs, they are hyper graphs.

这本书适合数学和计算机科学专业的学生，还有许多其他学科（比如生物信息学）的学生在学习图论、离散数学、数据结构和算法课程时阅读。它也适用于任何想了解图论的基本知识，或者只是对这有兴趣的人。不需要之前掌握图论或任何显著的数学知识。基本的集合论的知识、证明技巧以及算法都是理解这本书所需的条件；但这些概念都在文本中解释了。从结构上来说，本书被分为两部分，其中第二部分是第一部分的一般化。第一部分着重讨论了图论的关键概念，比如树、二分图、回路、嵌入图和图着色等。第二部分则考虑了第一部分的一般化，并探讨了超树、二分图超组、超回路、嵌入超图和超图着色等问题。各个部分之间以及各自的部分内部都展示了概念一般化的思路和交互方式。重点在于展示从最简单到最复杂的数学概念的推广方法及其互动。本书的一个特点是超图的概念具有对偶性。这一基本概念在图论（及其初级教学）中是缺失的，这是因为双顶点图不是真正的图，而是超图。

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