Planner Graph

Duration: 7 min

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Inside: a video lesson and guided study material.

Module outline

  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
  2. General Awareness: Physics, Chemistry, Biology, History, Geography, Polity, Indian Economy, Environment, Static General Knowledge, GK- Current Affairs, Artificial Intelligence, MS Office Applications, Terms & Abbreviations, Keyboard Shortcuts, Evolution of Indian Banking (Banking & Financial Awareness), Emerging Tech Trends
  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. 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, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: 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, Database Recovery, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. 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, Complexity Theory
  10. 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. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Security, Windows OS, Distributed Systems, Unix OS, Linux OS
  12. Computer Networks: Introduction to CN, 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, Data Communication, Website, Browser & Services, Web Design & Development, Network Security, Cyber Threats and Malware Attacks, Cyber Safety and Security
  13. Full Length Mock Tests:
  14. Sectional Tests:
  15. Paper 2 - Full-length Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video introduces planar graphs, defined as graphs drawable on a plane without edge crossings. The instructor uses a square with diagonals (K4) to show how a non-planar drawing can be transformed into a planar one. The lesson identifies K5 and K3,3 as fundamental non-planar graphs. Through visual demonstrations and mathematical derivations, the instructor proves their non-planarity using edge-vertex inequalities derived from Euler's formula. The lecture connects theoretical definitions with practical applications in engineering and circuit design.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a clear definition displayed on the screen: "Planer Graph: - A graph is called a planer graph if it can be drawn on a plan in such a way that no edges cross each other, otherwise it is called non-planer." The instructor highlights applications in civil engineering and circuit designing. He draws a square with vertices labeled a, b, c, and d, including both diagonals, which creates a crossing point in the center. He explains that while this drawing has crossings, the graph itself is planar. To prove this, he redraws the graph in red, keeping the outer square and one diagonal, but moving the other diagonal outside the square to avoid intersection. This visual transformation effectively demonstrates the core concept that planarity depends on the existence of a crossing-free drawing, not just any specific drawing. The labels a, b, c, and d are clearly visible at the corners of the square.

  2. 2:00 – 5:00 02:00-05:00

    The instructor shifts focus to non-planar graphs, introducing the complete graph K5. He draws a pentagon with all diagonals connected, forming a star inside, and labels it "K5". He attempts to redraw this graph to eliminate crossings but shows that it is impossible, marking it with a red cross. Next, he introduces the complete bipartite graph K3,3. He draws a complex graph with six vertices and nine edges, attempting to rearrange the edges to avoid intersections. He writes down the vertex and edge counts for these graphs: for K5, he writes "5" for vertices and "10" for edges; for K3,3, he writes "6" for vertices and "9" for edges. These visual aids serve as the basis for the mathematical proofs that follow, establishing K5 and K3,3 as the minimal non-planar graphs. The red drawings show the instructor's struggle to find a planar embedding.

  3. 5:00 – 7:15 05:00-07:15

    In the final segment, the instructor formalizes the non-planarity of K5 and K3,3 using mathematical inequalities. He writes the condition for a simple planar graph: E <= 3V - 6. He applies this to K5, substituting V=5 and E=10, resulting in the inequality 10 <= 9, which is false, thus proving K5 is non-planar. He then notes that for K3,3, the inequality 9 <= 12 holds true, so this condition alone is insufficient. He introduces a stricter condition for bipartite graphs: E <= 2V - 4. Applying this to K3,3 (V=6, E=9), he derives 9 <= 8, which is false. This mathematical proof confirms K3,3 is non-planar. The instructor concludes by summarizing that these two graphs are the fundamental obstructions to planarity. He also writes "min" under the inequalities to indicate minimum conditions.

The lecture effectively bridges the gap between visual intuition and mathematical rigor in graph theory. By starting with a simple definition and a tangible example of K4, the instructor makes the abstract concept of planarity accessible. The progression to K5 and K3,3 introduces the concept of forbidden minors. The final mathematical proofs provide the necessary tools for students to verify planarity without drawing, using edge and vertex counts. This structured approach ensures a deep understanding of why certain graphs are inherently non-planar.

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