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Computer Science

Graph Traversal Algorithms

BFS, DFS, and How Computers Explore Networks — A High School & College Primer

Graph traversal is one of those topics that shows up everywhere — AP Computer Science A, intro college CS courses, technical interviews — and yet most textbooks bury it under pages of theory before showing you a single concrete example. If you have an exam next week, a coding interview coming up, or you're just trying to make sense of BFS and DFS for the first time, this guide gets you there fast.

**TLDR: Graph Traversal Algorithms** covers everything a high school or early college student needs: what graphs are and how to talk about them, how to represent them in code with adjacency lists and matrices, and how both breadth-first search and depth-first search actually work — traced step by step on real examples, with pseudocode you can read and reuse. You'll see how BFS finds shortest paths in unweighted graphs, how DFS drives cycle detection and topological sort, and how to choose between the two when it matters. The final section connects these algorithms to real-world systems: social networks, web crawlers, GPS routing, and the path forward to Dijkstra and A*.

This is a 15-page primer, not a 600-page textbook. Every subsection leads with the one thing you need to take away, backs it up with worked examples and concrete numbers, and flags the misconceptions that trip students up most often. It's designed for a focused one- or two-hour study session — the kind of efficient, no-fluff prep that fits between classes or before a test.

If you need a clear, fast introduction to graph algorithms for your CS course or interview prep, start here.

What you'll learn
  • Define a graph in terms of vertices and edges and distinguish directed, undirected, weighted, and unweighted graphs.
  • Represent a graph in code using adjacency lists and adjacency matrices and choose the right one for a problem.
  • Trace and implement breadth-first search (BFS) using a queue, including shortest-path computation on unweighted graphs.
  • Trace and implement depth-first search (DFS) using recursion or a stack, and use it for cycle detection and connected components.
  • Compare BFS and DFS by time complexity, space complexity, and the kinds of problems each is best suited to solve.
What's inside
  1. 1. What Is a Graph?
    Introduces graphs as collections of vertices and edges, with the vocabulary and variants students will see throughout the book.
  2. 2. Representing Graphs in Code
    Compares adjacency lists and adjacency matrices, with code sketches and a discussion of when each is the right choice.
  3. 3. Breadth-First Search (BFS)
    Walks through BFS using a queue, traces it on a small graph, and shows how it computes shortest paths in unweighted graphs.
  4. 4. Depth-First Search (DFS)
    Develops DFS recursively and iteratively with a stack, traces it on the same graph, and introduces discovery/finish times.
  5. 5. BFS vs DFS: Choosing the Right Tool
    Side-by-side comparison of the two algorithms by complexity, traversal order, and the problems each one solves best.
  6. 6. Where Graph Traversal Shows Up
    Real applications — social networks, web crawlers, maze solving, GPS, and what to study next (Dijkstra, A*, etc.).
Published by Solid State Press
Graph Traversal Algorithms cover
TLDR STUDY GUIDES

Graph Traversal Algorithms

BFS, DFS, and How Computers Explore Networks — A High School & College Primer
Solid State Press

Graph Traversal Algorithms

BFS, DFS, and How Computers Explore Networks — A High School & College Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you are a high school student who needs a computer science algorithms review for an upcoming test, or you are working through AP CSA and want a focused data structures review that actually makes sense, this book is for you. It also fits any college freshman hitting graphs for the first time in an intro CS course, or anyone doing coding interview graph problems prep and needing to get up to speed fast.

This short guide to graphs and traversal in CS covers everything from the ground up: what graphs are, how to represent them in code, and how BFS and DFS work — with pseudocode, traced examples, and real applications. Think of it as a breadth-first and depth-first search tutorial paired with an intro to graphs computer science primer, all in about fifteen pages with zero filler.

Read it straight through once, then work every example yourself before checking the solution. Finish with the practice problems at the end to confirm you can apply what you have learned.

Contents

  1. 1 What Is a Graph?
  2. 2 Representing Graphs in Code
  3. 3 Breadth-First Search (BFS)
  4. 4 Depth-First Search (DFS)
  5. 5 BFS vs DFS: Choosing the Right Tool
  6. 6 Where Graph Traversal Shows Up
Chapter 1

What Is a Graph?

A graph is a way of representing relationships. You have a set of objects, and some pairs of those objects are connected. That's it. The objects are called vertices (singular: vertex), and the connections between them are called edges.

You already know several things that fit this description naturally. A map of cities and the roads between them is a graph. A social network where people are vertices and friendships are edges is a graph. The internet, where web pages are vertices and hyperlinks are edges, is a graph. Once you can see the pattern, you'll find graphs everywhere.

Formally, a graph $G$ is defined as a pair $(V, E)$, where $V$ is a set of vertices and $E$ is a set of edges. Each edge connects two vertices. If $V = \{A, B, C\}$ and $E = \{(A,B), (B,C)\}$, then you have three vertices with $A$ connected to $B$ and $B$ connected to $C$, but no direct connection between $A$ and $C$.

Directed vs. Undirected

The first major distinction is whether edges have direction. In an undirected graph, every edge goes both ways — if $A$ is connected to $B$, then $B$ is connected to $A$. Friendships on Facebook work this way: if you are my friend, I am your friend.

In a directed graph (also called a digraph), each edge has a specific direction, drawn as an arrow from one vertex to another. Following someone on Twitter is directed: you can follow someone without them following you back. When we write a directed edge, order matters — $(A, B)$ means there is an arrow from $A$ to $B$, but not necessarily one from $B$ to $A$.

Weighted vs. Unweighted

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon