Why Graph Terminology Vertices Edges Directed Undirected Weighted in DSA C#?

The Big Idea

Discover how graphs turn messy connections into clear maps you can explore and understand!

The Scenario

Imagine you want to map all your friends and how they are connected to each other. You try to write down every connection on paper, but soon it becomes messy and confusing.

You also want to know who is friends with whom, if the friendship is one-way or two-way, and if some friendships are stronger than others.

The Problem

Writing connections manually is slow and easy to mess up. You might forget some links or mix up directions.

It is hard to see the whole picture or answer questions like "Who is connected to whom?" or "Which friendships are stronger?"

The Solution

Graphs let you organize connections clearly using vertices (points) and edges (lines).

Vertices represent friends, edges represent friendships.

Directed edges show one-way connections, undirected edges show two-way.

Weighted edges add strength or cost to connections.

This makes it easy to store, explore, and analyze relationships.

Before vs After

✗ Before

string[] friends = {"Alice", "Bob", "Charlie"};
// Manually track connections with notes or lists
// No clear structure for direction or weight

✓ After

class Graph {
  Dictionary<string, List<(string, int)>> adjacencyList = new Dictionary<string, List<(string, int)>>();
  // Add vertices and edges with direction and weight
}

What It Enables

Graphs let you model and solve complex connection problems clearly and efficiently.

Real Life Example

Social networks use graphs to show who follows whom, who is friends with whom, and how strong those connections are.

Key Takeaways

Vertices are points or nodes representing entities.

Edges connect vertices and can be directed or undirected.

Weighted edges carry extra information like strength or cost.

Practice

1. In a graph, what do we call the points or nodes where edges meet?

easy

A. Vertices

B. Edges

C. Weights

D. Paths

Solution

1. Step 1: Understand graph components

   A graph consists of points called nodes and connections called edges.

2. Step 2: Identify the term for nodes

   The points or nodes where edges meet are called vertices.

3. Final Answer:

   Vertices -> Option A

4. Quick Check:

   Nodes = Vertices [OK]

Hint: Remember: nodes = vertices, connections = edges [OK]

Common Mistakes:

• Confusing edges with vertices
• Thinking weights are nodes
• Mixing paths with vertices

2. Which of the following is the correct way to declare a directed graph using adjacency list in C#?

easy

A. int[] graph = new int[10];

B. List<int> graph = new List<int>();

C. Dictionary graph = new Dictionary();

D. Dictionary graph = new Dictionary();

Solution

1. Step 1: Understand adjacency list representation

   An adjacency list uses a dictionary mapping each vertex to a list of connected vertices.

2. Step 2: Check the correct C# syntax

   Dictionary correctly represents a graph where each int key has a list of connected nodes.

3. Final Answer:

   Dictionary<int, List<int>> graph = new Dictionary<int, List<int>>(); -> Option C

4. Quick Check:

   Adjacency list = Dictionary with List values [OK]

Hint: Adjacency list uses Dictionary with List for edges [OK]

Common Mistakes:

• Using List<int> alone
• Using array instead of dictionary
• Using Dictionary which stores single values

3. Consider the following C# code snippet for an undirected weighted graph:

var graph = new Dictionary>();
graph[1] = new List<(int, int)> { (2, 5), (3, 10) };
graph[2] = new List<(int, int)> { (1, 5), (3, 3) };
graph[3] = new List<(int, int)> { (1, 10), (2, 3) };

// What is the weight of the edge between vertex 2 and vertex 3?

medium

A. 8

B. 5

C. 10

D. 3

Solution

1. Step 1: Locate edge from vertex 2 to 3

   In graph[2], the list contains (1,5) and (3,3). The tuple (3,3) means vertex 3 with weight 3.

2. Step 2: Identify the weight value

   The weight of the edge between 2 and 3 is 3.

3. Final Answer:

   3 -> Option D

4. Quick Check:

   Edge (2,3) weight = 3 [OK]

Hint: Check the tuple with target vertex for weight [OK]

Common Mistakes:

• Confusing vertex numbers
• Picking weight from wrong tuple
• Adding weights incorrectly

4. The following C# code tries to add an edge to an undirected graph but causes a runtime error:

var graph = new Dictionary();
graph[1] = new List<int> { 2 };
// Add edge from 2 to 1
graph[2].Add(1);

What is the cause of the error?

medium

A. graph[2] is not initialized before adding

B. Cannot add edges in undirected graphs

C. List<int> cannot be used for graph edges

D. Syntax error in dictionary declaration

Solution

1. Step 1: Check initialization of graph[2]

   graph[2] is not assigned any list before calling Add, so it is null.

2. Step 2: Understand why Add causes error

   Calling Add on null causes a NullReferenceException at runtime.

3. Final Answer:

   graph[2] is not initialized before adding -> Option A

4. Quick Check:

   Null list causes error [OK]

Hint: Always initialize lists before adding elements [OK]

Common Mistakes:

• Assuming dictionary auto-initializes lists
• Thinking undirected graphs disallow adding edges
• Confusing syntax error with runtime error

5. You have a weighted directed graph represented as:

var graph = new Dictionary>
{
  {1, new List<(int, int)>{ (2, 4), (3, 1) }},
  {2, new List<(int, int)>{ (3, 2) }},
  {3, new List<(int, int)>{ (4, 7) }},
  {4, new List<(int, int)>{}}
};

// What is the total weight of the path 1 -> 3 -> 4?

hard

A. 10

B. 8

C. 7

D. 3

Solution

1. Step 1: Find weight from 1 to 3

   From graph[1], edge to 3 has weight 1.

2. Step 2: Find weight from 3 to 4

   From graph[3], edge to 4 has weight 7.

3. Step 3: Calculate total path weight

   Total weight = 1 + 7 = 8.

4. Final Answer:

   8 -> Option B

5. Quick Check:

   1->3=1 + 3->4=7 = 8 [OK]

Hint: Sum weights along the path edges [OK]

Common Mistakes:

• Adding wrong edges
• Ignoring direction of edges
• Using wrong weights from graph