Maximumj independent line set

Duration: 3 min

This video lesson is available to enrolled students.

Return to /learn/COAL-INDIA-MANAGEMENT-TRAINEE/engineering-maths/graph-theory/graph-covering-line-and-vertex-covering-independent-set-and-minimalmaximal-variants/asset-maximumj-independent-line-set after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. Paper-2 | Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  3. Paper-2 | Databases / DBMS & SQL: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control
  4. Paper-2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  5. Paper-2 | Computer Networks & Security: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Network Access Technologies, Network Security
  6. Paper-2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  7. Paper-2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  8. Paper-2 | Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines
  9. Paper-2 | Computer Organization & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer
  10. Paper-2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  11. Paper-1 | General Awareness: Polity, Physics, Biology, Chemistry, Static General Knowledge, GK- Current Affairs, History, Geography, Indian Economy, Environment
  12. Paper-1 | Logical Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Data Interpretation, Blood Relations, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Clock, Calendar, Cubes & Dices, Directions (Direction Test), Analogy, Seating Arrangements, Analytical Reasoning (Counting Figures Reasoning)
  13. Paper-1 | Quantitative Aptitude: Ratio and Proportion (Ratios), Mixture and Alligation, Permutation and Combination, Number System, HCF LCM, Speed Time and Distance, Average, Age Problems, Percentage, Profit and Loss, Time and Work, Pipe and Cistern, Simple Interest and Compound Interest, Powers and Exponents (Surds and Indices), Mensuration and Geometry, Probability, Statistics, Arithmetic, Algebra
  14. Paper-1 | English: Vocabulary, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Cloze Test (Cloze Passage)
  15. Coal India Previous Year Paper:
  16. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus
  17. Coal India Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video defines an Independent Vertex Set as a subset of vertices where no two are adjacent. Using a graph with vertices a, b, c, d, e, the instructor validates S1={b}, S2={d, e}, and S3={a, c} as independent sets. He invalidates S4={b, e} because b and e are connected. This establishes the basic rule. The lecture then moves to Maximal and Maximum Independent Vertex Sets. A maximal set cannot have more vertices added. The maximum set has the largest number of vertices. S3={a, c, e} is shown as the maximum set with 3 vertices. The vertex independent number is denoted by beta_2. The formula alpha_2 + beta_2 = |V| relates this number to the total vertices. The instructor emphasizes these definitions with visual checks and crosses on the screen.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The first section defines an Independent Vertex Set, stating that no two vertices in the subset S can be adjacent. The instructor uses a graph with vertices a, b, c, d, e to demonstrate. Sets S1={b}, S2={d, e}, and S3={a, c} are marked with checks as valid independent sets. However, S4={b, e} is crossed out because b and e are adjacent, illustrating the core rule. The instructor explains that for a set to be independent, no edges can exist between its members. This practical demonstration reinforces the definition that no two vertices in the set can share an edge.

  2. 2:00 – 2:41 02:00-02:41

    The lecture defines Maximal and Maximum Independent Vertex Sets. A maximal set cannot have more vertices added. The maximum set has the most vertices. S3={a, c, e} is identified as the maximum set with size 3, denoted as beta_2(a)=3. The formula alpha_2 + beta_2 = |V| is circled, relating the independent number to the total vertices. The instructor explains these concepts while pointing to the lists on the screen. He notes that S1, S2, and S3 are maximal, but S3 is the maximum because it has the most vertices. The formula is highlighted to show the relationship between independent sets and the total vertex count.

The video progresses from basic definitions to advanced concepts in graph theory. It starts by establishing what an independent vertex set is through examples and non-examples. It then distinguishes between maximal and maximum sets, clarifying that maximal sets are locally optimal while maximum sets are globally optimal. The concept of the vertex independent number beta_2 is introduced as the size of the maximum set. Finally, the relationship alpha_2 + beta_2 = |V| is presented, linking the independent set size to the total graph size. This structured approach helps students understand the hierarchy of independent sets and their numerical properties. The instructor uses visual aids like checks and crosses to reinforce these abstract concepts effectively.

Loading lesson…