What is a Path in a Graph?

S3-SA5-0232

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A path in a graph is a sequence of connected vertices (nodes) where each vertex is linked to the next by an edge. Think of it as a route you can take from one point to another in a network.

Simple Example

Quick Example

Imagine a map of your city with different chai shops (vertices) and roads connecting them (edges). If you start at 'Sharma Ji ki Chai' and want to reach 'Gupta Ji ki Chai' by driving through 'Priya's Cafe', the sequence 'Sharma Ji -> Priya's Cafe -> Gupta Ji' forms a path.

Worked Example

Step-by-Step

Let's say we have a network of cities: Delhi (D), Mumbai (M), Kolkata (K), Chennai (C), Bengaluru (B). Roads connect D-M, D-K, M-B, K-C, B-C.

Step 1: Identify the starting city and the destination city. Let's find a path from Delhi (D) to Chennai (C).
---Step 2: Start at Delhi (D). What cities can you go to directly from D? Mumbai (M) and Kolkata (K).
---Step 3: Let's choose to go to Mumbai (M) first. So far, our path is D -> M.
---Step 4: From Mumbai (M), where can you go? Bengaluru (B). So now our path is D -> M -> B.
---Step 5: From Bengaluru (B), where can you go? Chennai (C). So now our path is D -> M -> B -> C.
---Step 6: We have reached Chennai (C), our destination. The path is complete.

Answer: One possible path from Delhi to Chennai is D -> M -> B -> C.

Why It Matters

Understanding paths is crucial for building smart systems that find the best routes, like Google Maps finding the fastest way to your school. It's used in computer science for network routing, in AI to solve problems, and even in economics to model trade routes, opening doors to careers in software development, logistics, and data analysis.

Common Mistakes

MISTAKE: Thinking an edge must be traversed only once in a path. | CORRECTION: While a 'simple path' doesn't repeat vertices or edges, a general 'path' can revisit vertices and edges, though usually, we look for simple paths.

MISTAKE: Confusing a path with a cycle. | CORRECTION: A path starts at one vertex and ends at another (or the same one), but a 'cycle' specifically means a path that starts and ends at the same vertex, visiting other vertices in between.

MISTAKE: Assuming all graphs have paths between any two vertices. | CORRECTION: A graph might be 'disconnected', meaning there's no path between certain pairs of vertices, just like some cities might not be reachable by road from others.

Practice Questions

Try It Yourself

QUESTION: In a graph with vertices A, B, C, D and edges A-B, B-C, C-D. Is A-B-C-D a path? | ANSWER: Yes, it is a path.

QUESTION: Given a graph with vertices P, Q, R, S and edges P-Q, Q-R, R-P. Find two different paths from P to R. | ANSWER: Path 1: P -> Q -> R. Path 2: P -> R (if there's a direct edge P-R, otherwise only Path 1). Assuming R-P means P-R is an edge, then P->R is a direct path too.

QUESTION: Consider a graph where your friends are vertices and a common interest is an edge. If you (A) like cricket with B, B likes movies with C, and C likes gaming with D, describe a path from A to D. | ANSWER: A -> B -> C -> D. (You share cricket with B, B shares movies with C, C shares gaming with D).

MCQ

Quick Quiz

Which of the following best describes a path in a graph?

A single vertex

A collection of unconnected edges

A sequence of connected vertices and edges

The total number of vertices in a graph

The Correct Answer Is:

A path is specifically a sequence where each vertex is connected to the next by an edge, forming a route. Options A, B, and D do not capture this sequential, connected nature.

Real World Connection

In the Real World

When you book a train ticket on IRCTC, the system uses graph theory to find available routes (paths) between your starting and destination stations, considering all the railway lines (edges) and stations (vertices) across India. Similarly, food delivery apps like Zomato or Swiggy find the shortest path for their delivery executives.

Key Vocabulary

Key Terms

VERTEX (NODE): A point or dot in a graph | EDGE: A line connecting two vertices | GRAPH: A collection of vertices and edges | SEQUENCE: An ordered list of items

What's Next

What to Learn Next

Now that you understand what a path is, you can explore 'Types of Paths' like simple paths and cycles, or learn about 'Shortest Path Algorithms'. These concepts build on understanding paths to solve more complex problems, like finding the quickest route!