Notion of connectivity and its importance in graph theory

The main source used in the preparation of this thesis is Diestel's (2017) "Graph Theory" book. Essentially, what we do is to translate the third chapter of the mentioned book into Turkish to understand and explain the concept of "connectivity in graph theory", as can be understood from the title of the thesis. But of course, no literal translation was made, on the one hand, some very difficult or technical proofs were skipped, on the other hand, some parts of the book that were left to the reader were explained and the subject was presented more understandably. In addition, the basic definitions and theorems of graph theory, which are prerequisites for understanding this study, are given by adding the necessary part of the first chapter of the aforementioned book to the thesis. In addition to these, the articles listed in the references were also consulted. To summarize: In the first chapter, after the basic definitions, necessary definitions and theorems related to roads and loops, connectedness, trees and forests, and bipartite graphs are covered. In the first two sections of the second chapter, the structures of 2-connected and 3-connected graphs are studied, respectively, and related definitions and theorems are given. In the third section, three different proofs of Menger's (1927) theorem, which is one of the most important theorems of graph theory, as well as the general form and proof of the theorem, are presented. In the fourth section, Mader's (1978) theorem, which is one of the most important and deep theorems of graph theory, which gives the number of independent paths that intersect an induced subgraph of a graph only at its endpoints, was stated, and proved. In the fifth and last section, the problem of the existence of discrete paths between two specified sets of vertices is briefly mentioned. In the third chapter of the thesis, before the conclusions and recommendation chapter, a brief literature review on the subject of the thesis is presented by using the notes section of the third chapter of the mentioned book.