--- title: Connectivity description: Connectivity measures the resilience of a graph when vertices or edges are removed. Learners will understand paths, cycles, cut-sets, and Menger's theorem, enabling them to analyze the reliability and flow of network infrastructures like communication and transportation systems. category: mathematics subcategory: graph-theory difficulty: beginner, intermediate, advanced url: /subject/connectivity --- # Connectivity Connectivity measures the resilience of a graph when vertices or edges are removed. Learners will understand paths, cycles, cut-sets, and Menger's theorem, enabling them to analyze the reliability and flow of network infrastructures like communication and transportation systems. ## Where to start If graphs are new to you, start with Introduction to Graph Theory on Coursera, a free beginner course needing only basic maths and some Python. If you only use one resource, make it Reinhard Diestel's Graph Theory, Chapter 3: Connectivity, released free by the author and covering Menger's theorem, 2- and 3-connected graphs and exercises. To compute connectivity with network flows, continue with Jeff Erickson's Applications of Flows and Cuts. ## Available Resources 1 Courses • 6 Websites ## Websites ### 1. Wikipedia: Connectivity (graph theory) Encyclopedia article defining vertex and edge connectivity, cut vertices, bridges, and k-connected graphs, stating Menger's theorem and bounds relating connectivity to minimum degree. Useful as a terminology reference with citations into the graph theory literature. **Difficulty:** Beginner | **Price:** Free **Link:** https://en.wikipedia.org/wiki/Connectivity_(graph_theory) **Tags:** graph-theory, connectivity, menger-theorem, vertex-cuts, discrete-mathematics ### 2. Diestel Graph Theory Official site for Reinhard Diestel's Springer graduate text, where the main text is readable free online and paid eBook editions add the full apparatus. It covers matching, connectivity, planarity, colouring, flows, extremal theory, and minors. **Difficulty:** Beginner | **Price:** Free **Link:** https://diestel-graph-theory.com/ **Tags:** graph-theory, textbook, discrete-mathematics, combinatorics, free-online ### 3. Applications of Flows and Cuts (Algorithms, chapter 11) **Author:** Jeff Erickson Answers the how-do-you-actually-compute-connectivity half of the topic note. Chapter from Erickson's free Illinois algorithms textbook whose opening sections reduce edge-disjoint and vertex-disjoint path counting to maximum flow, giving the constructive algorithmic side of Menger's theorem; later sections extend the machinery to matching. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://jeffe.cs.illinois.edu/teaching/algorithms/book/11-maxflowapps.pdf **Tags:** graph-theory, network-flow, max-flow-min-cut, menger-theorem, algorithms ### 4. Menger's Theorems and the Ear Lemma (Charles University, Lecture 8) **Author:** Irena Penev Lecture handout from Charles University's NDMI011 course containing a complete induction proof of the vertex form of Menger's theorem, its edge and global variants, and the ear lemma characterising 2-connected graphs. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://iuuk.mff.cuni.cz/~ipenev/KG1W2021Lecture08.pdf **Tags:** graph-theory, menger-theorem, 2-connected-graphs, ear-decomposition, proofs ### 5. Graph Theory, Chapter 3: Connectivity (Diestel, 6th edition) **Author:** Reinhard Diestel The complete Connectivity chapter of the sixth edition, released free by the author: 2-connected graphs and blocks, the structure of 3-connected graphs, Menger's theorem, Mader's theorem, edge-disjoint spanning trees, linking pairs of vertices, plus exercises. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.math.uni-hamburg.de/home/diestel/books/graph.theory/preview/Ch3.pdf **Tags:** graph-theory, connectivity, menger-theorem, 3-connected-graphs, textbook ### 6. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Youtubes ### 1. Graph Theory Lecture 9: Connectivity II — Menger's Theorem and Network Flows **Author:** Reinhard Diestel Ninety-nine-minute lecture recorded live at Hamburg University in 2023/24, following the sixth edition of Diestel's book: proofs of the vertex and edge forms of Menger's theorem, global versions, and the max-flow min-cut theorem. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=6Qky-jrNV1E **Tags:** graph-theory, menger-theorem, max-flow-min-cut, connectivity, lecture ## Courses ### 1. Introduction to Graph Theory We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. In this online course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible! By the end of the course, we will implement an algorithm which finds an optimal assignment of students to schools. This algorithm, developed by David Gale and Lloyd S. Shapley, was later recognized by the conferral of Nobel Prize in Economics. As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students. **Difficulty:** Beginner | **Language:** English | **Duration:** 5 weeks, 3-5 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/graphs **Tags:** courses, mathematics-statistics, combinatorics --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/connectivity