Graph theory objects | Computational problems in graph theory
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property. For example, in the graph P3, a path with three vertices a, b, and c, and two edges ab and bc, the sets {b} and {a, c} are both maximally independent. The set {a} is independent, but is not maximal independent, because it is a subset of the larger independent set {a, c}. In this same graph, the maximal cliques are the sets {a, b} and {b, c}. A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets. A graph may have many MISs of widely varying sizes; the largest, or possibly several equally large, MISs of a graph is called a maximum independent set. The graphs in which all maximal independent sets have the same size are called well-covered graphs. The phrase "maximal independent set" is also used to describe maximal subsets of independent elements in mathematical structures other than graphs, and in particular in vector spaces and matroids. Two algorithmic problems are associated with MISs: and . (Wikipedia).
Independent Vertex Sets | Graph Theory, Maximal and Maximum Independent Sets
What are independent vertex sets in graph theory? We'll go over independent sets, their definition and examples, and some related concepts in today's video graph theory lesson! A subset of the vertex set of a graph is an independent vertex set if and only if it contains no pair of adjace
From playlist Set Theory
Every Subset of a Linearly Independent Set is also Linearly Independent Proof
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys A proof that every subset of a linearly independent set is also linearly independent.
From playlist Proofs
Complement of Independent Set is Vertex Cover | Graph Theory
We prove the complement of an independent vertex set is a vertex cover. This makes for an easy direct proof once we recall our definitions. An independent vertex set is a set of vertices, no two of which are adjacent. A vertex cover is a set of vertices such that every edge has at least on
From playlist Graph Theory
Find a Set with Greatest Cardinality that is a Subset of Two Given Sets (Lists)
This video explains how to determine a set with greatest cardinality that is a subset of two given sets.
From playlist Sets (Discrete Math)
How to Find a Minimal Generating Set
How to Find a Minimal Generating Set
From playlist Linear Algebra
What are Overlapping Sets? | Set Theory
What are overlapping sets? This is a relation between sets that I have not seen any YouTube videos on, so I figured I'd add this video explaining the term to the massive YouTube catalogue! In this video we define overlapping sets and give some examples. Two sets, A and B, are overlapping
From playlist Set Theory
Find a Set with Least Cardinality that has Two Given Subsets (Lists)
This video explains how to determine a set with least cardinality that has two given subsets.
From playlist Sets (Discrete Math)
Is the Empty Set an Independent Vertex Set? | Graph Theory Exercises
We prove the empty set is an independent set using the definition of independent vertex sets. We also show the empty set is an independent set by considering the complement of a vertex cover. Recall that an independent vertex set S of a graph G is a subset of the vertex set of G such that
From playlist Graph Theory Exercises
Why every vector space (not necessarily finite dimensional) has a basis, feat. Zorn's Lemma and the actual definition of a basis Check out my set theory playlist: Set theory https://www.youtube.com/playlist?list=PLJb1qAQIrmmDr_RYAtqY1MNgTgVNMJNIf Check out my vector space playlist: https
From playlist Vector Spaces
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
Graph Theory: 50. Maximum vs Maximal
Here we describe the difference between two similar sounding words in mathematics: maximum and maximal. We use concepts in graph theory to highlight the difference. In particular, we define an independent set in a graph and a component in a graph and look at some examples. -- Bits of Gra
From playlist Graph Theory part-9
35 - Properties of bases (continued)
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
Lecture 5: Zorn’s Lemma and the Hahn-Banach Theorem
MIT 18.102 Introduction to Functional Analysis, Spring 2021 Instructor: Dr. Casey Rodriguez View the complete course: https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ YouTube Playlist: https://www.youtube.com/watch?v=KlAjiDivJoQ&list=PLUl4u3cNGP63micsJp_
From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021
Nexus Trimester - Ofer Shayevitz (Tel Aviv University)
Zero-error capacity for multiuser channels Ofer Shayevitz (Tel Aviv University) March,03 206 Abstract: The capacity of a point-to-point communication channel under a zero-error criterion was originally studied by Shannon in 1956. Despite the apparent simplicity of the problem, and in cont
From playlist Nexus Trimester - 2016 - Central Workshop
Matchings, Perfect Matchings, Maximum Matchings, and More! | Independent Edge Sets, Graph Theory
What are matchings, perfect matchings, complete matchings, maximal matchings, maximum matchings, and independent edge sets in graph theory? We'll be answering that great number of questions in today's graph theory video lesson! A matching in a graph is a set of edges with no common end-ve
From playlist Graph Theory
Deep InfoMax: Learning deep representations by mutual information estimation and maximization | AISC
For more details including paper and slides, visit https://aisc.a-i.science/events/2019-04-11/ Discussion lead/coauthor: Karan Grewal Abstract Building agents to interact with the web would allow for significant improvements in knowledge understanding and representation learning. Howev
From playlist Natural Language Processing
What is a Power Set? | Set Theory, Subsets, Cardinality
What is a power set? A power set of any set A is the set containing all subsets of the given set A. For example, if we have the set A = {1, 2, 3}. Then the power set of A, denoted P(A), is {{ }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} where { } is the empty set. We also know that
From playlist Set Theory
Abstract Algebra | Maximal and prime ideals.
We prove some classic results involving maximal and prime ideals. Specifically we prove the an ideal P is prime iff R/P is an integral domain. Further, we prove that an ideal M is maximal iff R/M is a field. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 ht
From playlist Abstract Algebra
Nonlinear algebra, Lecture 13: "Polytopes and Matroids ", by Mateusz Michalek
This is the thirteenth lecture in the IMPRS Ringvorlesung, the advanced graduate course at the Max Planck Institute for Mathematics in the Sciences.
From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra