M1_VOL3_GRAPHTHEORY 📈

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Here is a structured, detailed explanation of all major concepts in the PDF M1_VOL3_GRAPHTHEORY.pdf, including definitions, examples, related questions, and answers—all presented for easy understanding¹.

1. Graphs and General Graph Problems

1.1 Introduction

Concept: Graphs model pairwise relationships between objects.
Example: Social networks (people as vertices, friendships as edges), communication networks (devices as vertices, links as edges)¹.
Key Idea: Graphs abstract real-world situations by focusing on connections rather than physical layout.

1.2 Graph

Definition: A graph $ G = (V, E) $ consists of a set of vertices (nodes) $ V $ and a set of edges $ E $ connecting pairs of vertices.
Example: $ V = {A, B, C, D, E, F, G} $, $ E = {(A,B), (A,C), (B,D), (B,E), (C,F), (C,G)} $¹.
Undirected Graph: Edges have no direction; if $(A, B)$ is present, so is $(B, A)$ implicitly.

1.3 Types of Graphs

Simple Graph: No loops or multiple edges between the same pair of vertices.
Directed Graph: Edges have direction; $(A, B)$ does not imply $(B, A)$.
Undirected Graph: All edges are bidirectional.
Complete Graph: Every pair of distinct vertices is connected by an edge.
Example: A complete graph with 4 vertices has every vertex connected to every other vertex.

1.4 Paths and Reachability

Path: A sequence of vertices connected by edges.
Example: $ A \rightarrow B \rightarrow C \rightarrow D $ is a path from $ A $ to $ D $¹.
Reachability: Vertex $ u $ is reachable from $ v $ if there is a path from $ v $ to $ u $.
Example: In a social network, if Alice is connected to Bob, who is connected to Charlie, Alice is reachable from Charlie via Bob.

1.5 More on Graphs

1.5.1 Graph Coloring

Definition: Assign colors to vertices so that no two adjacent vertices have the same color.
Chromatic Number: Minimum number of colors needed.
Example: Scheduling classes so that conflicting classes (edges) are not at the same time (color). Result: 2 colors may suffice for some graphs¹.
Related Question: What is the minimum number of colors required for a given graph?
Answer: Depends on the graph; for the example in the PDF, it is 2.

1.5.2 Vertex Cover

Definition: A set of vertices such that every edge is incident to at least one vertex in the set.
Example: In a graph, ${2,4,5}$ may be a vertex cover¹.
Related Question: Find a vertex cover for a given graph.
Answer: For the example, ${2,4,5}$ is a vertex cover.

1.5.3 Independent Set

Definition: A set of vertices where no two are adjacent.
Example: ${1,4,6}$ may be a maximum independent set¹.
Related Question: Find a maximum independent set.
Answer: For the example, ${1,4,6}$ is a maximum independent set.

1.5.4 Matching

Definition: A set of edges without common vertices.
Example: ${(1,2), (3,4), (5,6)}$ is a matching¹.
Related Question: Find a maximum matching.
Answer: For the example, ${(1,2), (3,4), (5,6)}$ is a maximum matching.

1.6 Representing Graphs

1.6.1 Adjacency Matrix

Definition: A square matrix where $ A_{ij} = 1 $ if there is an edge between vertices $ i $ and $ j $, else 0.
Example:

$$ A = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{pmatrix} $$

Related Question: Find the adjacency matrix for a given graph.
Answer: See above matrix for the example.

1.6.2 Adjacency List

Definition: For each vertex, list its neighbors.
Example:
- $ A: {B} $
- $ B: {A, C, D, E} $
- $ C: {B, D} $
- $ D: {B, C, E} $
- $ E: {B, D} $

1.7 Breadth-First Search (BFS)

Algorithm: Explore all neighbors of a vertex before moving to the next level.
Example: Starting from vertex 1, BFS visits: 1, 2, 4, 3, 5, 6, 7¹.
Applications: Shortest path in unweighted graphs.
Related Question: Draw the BFS tree starting from vertex $ E $.
Answer: The tree will show $ E $ connected to its neighbors, then their neighbors, etc.

1.8 Depth-First Search (DFS)

Algorithm: Explore as far as possible along each branch before backtracking.
Example: Starting from vertex 4, DFS may visit: 4, 0, 1, 2, 3¹.
Applications: Topological sorting, strongly connected components, maze solving.
Related Question: Draw the DFS tree starting from vertex $ E $.
Answer: The tree will show a path as deep as possible before backtracking.

1.9 Degree of a Vertex

Definition: Number of edges incident to a vertex (undirected graph).
Example: In a complete graph with 4 vertices, each vertex has degree 3.
Related Question: What is the degree of each vertex in a given graph?
Answer: For the complete graph, all degrees are 3.

1.10 Indegrees and Outdegrees

Indegree: Number of edges entering a vertex (directed graph).
Outdegree: Number of edges leaving a vertex (directed graph).
Example: For a directed graph, sum of indegrees equals sum of outdegrees.
Related Question: What are the indegree and outdegree of each vertex?
Answer: For the example, indegree sequence is (1,1,1,0), outdegree is (1,2,1,0).

1.11 Problems

Example Question: Find the shortest path connecting two people in a social network.
Answer: Use BFS to find the shortest path.
More Questions: Find adjacency matrix, vertex cover, independent set, BFS/DFS trees, chromatic number, etc.

2. DAGs, Topological Sorting, and Longest Path

2.1 Directed Acyclic Graph (DAG)

Definition: Directed graph with no directed cycles.
Example: Task dependencies in project scheduling.
Related Question: Why is a given graph not a DAG?
Answer: Because it contains a directed cycle.

2.2 Topological Sorting

Definition: Linear ordering of vertices such that for every directed edge $(u, v)$, $u$ comes before $v$.
Algorithm: Repeatedly pick vertices with indegree 0, remove them, and update indegrees.
Example: For a DAG, one possible topological order is $A, B, D, E, C, F$¹.
Related Question: Find a topological sorting for a given DAG.
Answer: $A, B, D, E, C, F$ is one possible order.

2.3 Longest Path in a DAG

Algorithm: Topologically sort the graph, then for each vertex, update the longest path to its neighbors.
Example: In a DAG, the longest path can be found using dynamic programming after topological sort¹.

2.4 Transitive Closure

Definition: A graph that includes an edge $(u, v)$ if there is a path from $u$ to $v$ in the original graph.
Example: If there are paths $A \rightarrow B \rightarrow C$, then the transitive closure includes $A \rightarrow C$¹.
Related Question: Find the transitive closure of a given graph.
Answer: Add edges for all reachable pairs.

2.5 Matrix Multiplication

Adjacency Matrix: Represents graph connectivity.
Reachability Matrix: $A^k$ gives paths of length $k$.
Transitive Closure Matrix: $A + A^2 + A^3 + \ldots + A^n$.
Example: For a graph, compute $A^2$ to find paths of length 2¹.
Related Question: Compute $A^2$ for a given adjacency matrix.
Answer: Multiply the matrix by itself.

2.6 Problems

Example Question: Which relation represents the transitive closure?
Answer: The relation that includes all reachable pairs.
More Questions: Find matrix powers, topological sorting, longest path, etc.

3. Weighted Graphs and Shortest Path Algorithms

3.1 Weighted Graph

Definition: Each edge has a weight (distance, cost, time).
Example: Cities connected by roads with distances¹.

3.2 Dijkstra’s Algorithm

Algorithm: Finds shortest path from a source to all other vertices in a graph with non-negative weights.
Example: Shortest path from $A$ to $D$: $A \rightarrow C \rightarrow D$ with total weight 3¹.
Related Question: Find the shortest path from $A$ to $D$.
Answer: $A \rightarrow C \rightarrow D$ with weight 3.

3.3 Bellman-Ford Algorithm

Algorithm: Finds shortest paths from a source in graphs with negative weights (no negative cycles).
Example: After iterations, shortest distances from $A$: $A(0), B(-1), C(2), D(1), E(4)$¹.
Related Question: What are the shortest distances after Bellman-Ford?
Answer: As above.

3.4 Spanning Trees

Definition: A subgraph that is a tree and connects all vertices.
Example: A tree connecting all cities with minimum total road length¹.

3.5 Prim’s Algorithm

Algorithm: Greedily adds the shortest edge connecting a tree vertex to a non-tree vertex.
Example: Starting from $A$, add edges $(A,C)$, $(C,E)$, etc., to form a minimum spanning tree¹.
Related Question: Find the minimum spanning tree using Prim’s algorithm.
Answer: Add edges in order of smallest weight, avoiding cycles.

3.6 Kruskal’s Algorithm

Algorithm: Adds edges in order of increasing weight, skipping those that form cycles.
Example: Add edges $(B,D)$, $(A,C)$, $(A,F)$, etc., to form a minimum spanning tree¹.
Related Question: Find the minimum spanning tree using Kruskal’s algorithm.
Answer: Add edges in order of increasing weight, skipping those that form cycles.

3.7 Problems

Example Question: At what time will city $G$ start flooding if water spreads along weighted edges?
Answer: 8 minutes¹.
More Questions: Find shortest paths, minimum spanning trees, order of edge addition, etc.

4. Answers to Selected Questions

Graph Coloring: Minimum number of colors is 2 for the example graph.
Vertex Cover: ${2,4,5}$ is a vertex cover.
Independent Set: ${1,4,6}$ is a maximum independent set.
Matching: ${(1,2), (3,4), (5,6)}$ is a maximum matching.
BFS/DFS Trees: See above for examples.
Degree/Indegree/Outdegree: See above for examples.
Topological Sorting: $A, B, D, E, C, F$ is one possible order.
Transitive Closure: Add edges for all reachable pairs.
Dijkstra’s Algorithm: Shortest path from $A$ to $D$ is $A \rightarrow C \rightarrow D$ with weight 3.
Bellman-Ford: Shortest distances from $A$ are $A(0), B(-1), C(2), D(1), E(4)$.
Prim’s/Kruskal’s: Add edges in order of smallest weight, avoiding cycles.

Summary Table

Concept	Definition/Algorithm	Example/Question	Answer/Explanation
Graph	$G = (V, E)$	Social network	Vertices: people, edges: friendships
Path/Reachability	Sequence of connected vertices	$A \rightarrow B \rightarrow C$	Path from $A$ to $C$
Graph Coloring	Color vertices, no two adjacent same	Scheduling classes	2 colors for example graph
Vertex Cover	Set covers all edges	${2,4,5}$	Covers all edges
Independent Set	No two vertices adjacent	${1,4,6}$	Maximum independent set
Matching	No two edges share a vertex	${(1,2), (3,4), (5,6)}$	Maximum matching
Adjacency Matrix	$A_{ij} = 1$ if edge $i-j$	See matrix above	Represents graph connectivity
BFS	Explore level by level	1, 2, 4, 3, 5, 6, 7	Shortest path in unweighted
DFS	Explore as deep as possible	4, 0, 1, 2, 3	Topological sorting, etc.
Degree	Number of edges at vertex	3 in complete 4-vertex graph	All degrees 3
Indegree/Outdegree	Edges in/out (directed graph)	Indegree: (1,1,1,0), Outdegree: (1,2,1,0)	Sums equal
DAG	No directed cycles	Task dependencies	No cycles
Topological Sorting	Linear order, edges $u \rightarrow v$, $u$ before $v$	$A, B, D, E, C, F$	One possible order
Longest Path (DAG)	Dynamic programming after topo sort	See example	Longest path found
Transitive Closure	Add edges for all reachable pairs	$A \rightarrow C$ if $A \rightarrow B \rightarrow C$	All reachable pairs
Matrix Multiplication	$A^k$ gives paths of length $k$	See matrix multiplication	Paths of length $k$
Dijkstra’s Algorithm	Shortest path, non-negative weights	$A \rightarrow C \rightarrow D$	Shortest path, weight 3
Bellman-Ford	Shortest path, negative weights	$A(0), B(-1), C(2), D(1), E(4)$	Shortest distances
Spanning Tree	Tree connecting all vertices	See example	Connects all, no cycles
Prim’s Algorithm	Greedy, add smallest edge	$(A,C), (C,E), \ldots$	Minimum spanning tree
Kruskal’s Algorithm	Add edges in order, avoid cycles	$(B,D), (A,C), \ldots$	Minimum spanning tree

This structured approach covers all major concepts in the graph theory PDF, with clear definitions, examples, and answers to related questions¹.

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M1_VOL2_CALCULUS.pdf Mathematics I 🔥