Why Graph Terminology Vertices Edges Directed Undirected Weighted in DSA Typescript?

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 friendship and direction manually on paper.

The Problem

Writing and updating connections by hand is slow and confusing. You might forget who is connected to whom or miss if the connection has a direction or weight, like how close the friendship is.

The Solution

Graphs let you organize all these connections clearly. You can represent friends as points (vertices) and friendships as lines (edges). You can also show if friendships go one way (directed) or both ways (undirected), and add importance (weights) easily.

Before vs After

✗ Before

const friends = ['Alice', 'Bob', 'Carol'];
// Manually track connections with notes
// Alice -> Bob (close), Bob <-> Carol (friends)

✓ After

class Graph {
  vertices: string[] = [];
  edges: Map<string, Map<string, number>> = new Map();
  // Add vertices and edges with direction and weight
}

What It Enables

Graphs let you quickly find relationships, paths, and strengths between points in complex networks.

Real Life Example

Social media platforms use graphs to show who follows whom, suggest friends, and rank connections by interaction strength.

Key Takeaways

Vertices are points or nodes representing entities.

Edges connect vertices and can have direction and weight.

Graphs help model complex relationships clearly and efficiently.

Practice

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

easy

A. Edges

B. Weights

C. Vertices

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 C

4. Quick Check:

   Nodes = Vertices [OK]

Hint: Nodes in graphs are called vertices [OK]

Common Mistakes:

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

2. Which of the following is the correct way to represent a weighted edge from vertex 1 to vertex 2 with weight 5 in TypeScript?

easy

A. const edge = { 1: 2, 5 };

B. const edge = [1, 2, 5];

C. const edge = (1, 2, 5);

D. const edge = { from: 1, to: 2, weight: 5 };

Solution

1. Step 1: Understand object syntax in TypeScript

   Objects use key-value pairs with curly braces and colons.

2. Step 2: Check each option's syntax

   const edge = { from: 1, to: 2, weight: 5 }; correctly uses keys 'from', 'to', and 'weight' with values. const edge = [1, 2, 5]; is an array, valid but less descriptive. const edge = (1, 2, 5); uses parentheses, invalid for object. const edge = { 1: 2, 5 }; has invalid syntax.

3. Final Answer:

   const edge = { from: 1, to: 2, weight: 5 }; -> Option D

4. Quick Check:

   Object with keys and values = const edge = { from: 1, to: 2, weight: 5 }; [OK]

Hint: Use curly braces and key:value pairs for objects [OK]

Common Mistakes:

• Using parentheses instead of braces
• Incorrect object key syntax
• Confusing arrays with objects

3. What will be the output of the following TypeScript code that creates a directed weighted graph and prints its edges?

type Edge = { from: number; to: number; weight: number };
const graph: Edge[] = [
  { from: 1, to: 2, weight: 3 },
  { from: 2, to: 3, weight: 4 },
  { from: 3, to: 1, weight: 5 }
];
for (const edge of graph) {
  console.log(`${edge.from} -> ${edge.to} (weight: ${edge.weight})`);
}

medium

A. 1 -> 2 (weight: 3)
2 -> 3 (weight: 4)
3 -> 1 (weight: 5)

B. 1 -> 2 (weight: 5)
2 -> 3 (weight: 3)
3 -> 1 (weight: 4)

C. 1 -> 3 (weight: 5)
2 -> 1 (weight: 4)
3 -> 2 (weight: 3)

D. Error: Invalid graph definition

Solution

1. Step 1: Read the graph array

   The graph array has edges: 1->2 with weight 3, 2->3 with weight 4, 3->1 with weight 5.

2. Step 2: Trace the for loop output

   The loop prints each edge in the format 'from -> to (weight: weight)'. So output matches 1 -> 2 (weight: 3) 2 -> 3 (weight: 4) 3 -> 1 (weight: 5) exactly.

3. Final Answer:

   1 -> 2 (weight: 3) 2 -> 3 (weight: 4) 3 -> 1 (weight: 5) -> Option A

4. Quick Check:

   Print edges as defined = 1 -> 2 (weight: 3) 2 -> 3 (weight: 4) 3 -> 1 (weight: 5) [OK]

Hint: Match printed edges exactly as in array order [OK]

Common Mistakes:

• Mixing up weights and vertices
• Reversing edge directions
• Assuming syntax error

4. Identify the error in the following TypeScript code that tries to create an undirected weighted graph:

type Edge = { from: number; to: number; weight: number };
const graph: Edge[] = [
  { from: 1, to: 2, weight: 7 },
  { from: 2, to: 1, weight: 7 },
  { from: 2, to: 3, weight: 5 }
];
function addEdge(from: number, to: number, weight: number) {
  graph.push({ from, to, weight });
  graph.push({ from: to, to: from, weight });
}
addEdge(3, 4, 6);
console.log(graph.length);

medium

A. No error; graph length will be 5

B. Error: Duplicate edges cause conflict

C. Error: addEdge function pushes edges incorrectly

D. Error: graph array is immutable

Solution

1. Step 1: Count initial edges

   Initially, graph has 3 edges: 1->2, 2->1, 2->3.

2. Step 2: Analyze addEdge function call

   addEdge(3, 4, 6) pushes { from: 3, to: 4, weight: 6 } and { from: 4, to: 3, weight: 6 }, adding 2 more edges for undirected graph.

3. Step 3: Calculate final length

   3 initial + 2 added = 5. No errors occur; code runs successfully.

4. Final Answer:

   No error; graph length will be 5 -> Option A

5. Quick Check:

   3 + 2 = 5 [OK]

Hint: Count edges before and after addEdge call [OK]

Common Mistakes:

• Assuming duplicate edges cause error
• Miscounting edges after push
• Thinking graph is immutable

5. You have a weighted undirected graph represented as an adjacency list in TypeScript:

const graph: Record = {
  1: [{ node: 2, weight: 4 }, { node: 3, weight: 1 }],
  2: [{ node: 1, weight: 4 }, { node: 3, weight: 2 }, { node: 4, weight: 5 }],
  3: [{ node: 1, weight: 1 }, { node: 2, weight: 2 }, { node: 4, weight: 8 }],
  4: [{ node: 2, weight: 5 }, { node: 3, weight: 8 }]
};

// How to find the total weight of edges connected to vertex 2?

Which code snippet correctly calculates the total weight of edges connected to vertex 2?

hard

A. const total = graph[2].map(edge => edge.weight).length;

B. const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0);

C. const total = graph[2].filter(edge => edge.weight > 0);

D. const total = graph.reduce((sum, edges) => sum + edges.weight, 0);

Solution

1. Step 1: Understand adjacency list structure

   graph[2] is an array of edges connected to vertex 2, each with a weight property.

2. Step 2: Calculate total weight

   Use reduce to sum all edge weights in graph[2]. const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0); correctly sums weights. const total = graph[2].map(edge => edge.weight).length; counts edges, not sum. const total = graph[2].filter(edge => edge.weight > 0); filters edges, no sum. const total = graph.reduce((sum, edges) => sum + edges.weight, 0); tries to reduce graph object, invalid.

3. Final Answer:

   const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0); -> Option B

4. Quick Check:

   Sum weights with reduce = const total = graph[2].reduce((sum, edge) => sum + edge.weight, 0); [OK]

Hint: Use reduce to sum weights in adjacency list [OK]

Common Mistakes:

• Using map length instead of sum
• Filtering without summing
• Reducing wrong data structure