Computer Science · 10th Grade · Algorithmic Logic and Complexity · Weeks 1-9

Introduction to Graph Algorithms

Students are introduced to basic graph theory concepts and simple graph traversal algorithms.

• Common Core State Standards
• CSTA: 3A-AP-14

About This Topic

Graphs are one of the most broadly applicable data structures in computer science. Social networks, road systems, web links, dependency trees, and circuit layouts are all naturally modeled as graphs. In 10th grade, students encounter graphs at the point where their algorithmic toolkit is rich enough to apply it to these structures. CSTA standard 3A-AP-14 explicitly calls for students to use data structures and algorithms together to solve problems, and graphs are a natural synthesis of both.

The core concepts are accessible: a graph consists of vertices (nodes) and edges (connections between them). Breadth-First Search (BFS) explores a graph layer by layer, visiting all nodes at distance 1 before distance 2, making it useful for finding shortest paths. Depth-First Search (DFS) follows one path as far as possible before backtracking, making it useful for exploring all reachable nodes or detecting cycles. Both algorithms are building blocks for more advanced graph problems in future courses.

Active learning is well-suited to graph algorithms because graphs are inherently visual and social. Physical simulations where students are nodes and connections between them are edges make traversal behavior immediate and memorable.

Key Questions

1. Explain the components of a graph and their representation.
2. Analyze the process of Breadth-First Search (BFS) on a given graph.
3. Predict the path taken by Depth-First Search (DFS) from a starting node.

Learning Objectives

• Identify the components of a graph, including vertices and edges, from a given diagram or description.
• Explain the step-by-step process of Breadth-First Search (BFS) traversal on a provided graph.
• Predict the sequence of visited nodes when applying Depth-First Search (DFS) starting from a specified node in a graph.
• Compare the traversal paths generated by BFS and DFS on the same graph, highlighting key differences in exploration strategy.

Before You Start

Introduction to Data Structures

Why: Students need a foundational understanding of how data can be organized to grasp the concept of vertices and edges as components of a data structure.

Basic Algorithmic Thinking

Why: Familiarity with sequential execution, loops, and conditional statements is necessary to understand the logic of traversal algorithms.

Key Vocabulary

Vertex (Node)	A fundamental unit in a graph, representing an object or point. Multiple vertices are connected by edges.
Edge	A connection between two vertices in a graph. Edges can be directed or undirected, representing relationships or links.
Graph Traversal	The process of visiting each vertex in a graph systematically. Algorithms like BFS and DFS are common traversal methods.
Breadth-First Search (BFS)	A graph traversal algorithm that explores neighbor nodes first before moving to the next level neighbors. It is often used to find the shortest path in an unweighted graph.
Depth-First Search (DFS)	A graph traversal algorithm that explores as far as possible along each branch before backtracking. It is useful for finding cycles or exploring all reachable nodes.

Watch Out for These Misconceptions

Common Misconception

BFS and DFS visit the same nodes in the same order.

What to Teach Instead

Both algorithms visit all reachable nodes, but in very different orders. BFS visits by distance from the start (nearest first), while DFS follows one path as deep as possible before backtracking. For finding the shortest path in an unweighted graph, BFS is correct; DFS finds a path but not necessarily the shortest one. Physical simulations make the difference vivid.

Common Misconception

A graph is just a more complicated list or array.

What to Teach Instead

Lists and arrays are linear structures; graphs represent arbitrary connections between any nodes, including cycles. The jump from sequential to relational structure is significant. Students who treat graphs as linear often struggle with the need to track visited nodes, which is a key part of both BFS and DFS and a requirement for correctness on graphs with cycles.

Active Learning Ideas

Role Play: Human BFS and DFS

Students each receive a node number. For BFS, the teacher calls out nodes by layer: start node, then all its neighbors, then all their unvisited neighbors. For DFS, one path is followed as deep as possible before backtracking. Students stand when visited. The contrast in visiting order between BFS and DFS is immediate and memorable with 10-15 students.

25 min·Whole Class

Inquiry Circle: Trace the Traversal

Groups of 3 receive a drawn graph with 7-8 nodes and perform both BFS and DFS starting from the same node, writing out the order of visited nodes for each. Groups compare their BFS and DFS orderings and explain why they differ. Surfaces that the same graph yields different valid orderings and that algorithm choice determines what you find first.

35 min·Small Groups

Think-Pair-Share: Which Search Fits the Problem?

Present four real-world scenarios: finding the shortest path in a road network, checking if two users are connected in a social network, exploring all files reachable from a root directory, and detecting circular dependencies in software packages. Pairs match each to BFS or DFS and justify the choice, then share reasoning with the class.

25 min·Pairs

Gallery Walk: Graph Representation Challenge

Post four graphs around the room, each shown in one format: adjacency matrix, adjacency list, edge list, or visual diagram. Students must create the other three representations for each graph they visit. Reinforces that the same graph has multiple valid representations and that algorithm implementation depends on knowing which to use.

40 min·Small Groups

Real-World Connections

• Google Maps uses graph algorithms to find the shortest or fastest routes between locations, representing roads as edges and intersections as vertices.
• Social media platforms like Facebook model user connections as a graph, where users are vertices and friendships are edges, enabling features like friend suggestions and network analysis.
• Network engineers use graph traversal algorithms to troubleshoot connectivity issues and optimize data flow across computer networks, treating routers and devices as vertices and connections as edges.

Assessment Ideas

Quick Check

Provide students with a small, labeled graph. Ask them to draw the order in which nodes would be visited using BFS starting from a specific node, and then repeat for DFS. Check for correct application of each algorithm's rules.

Exit Ticket

On an index card, have students define 'vertex' and 'edge' in their own words. Then, ask them to describe one scenario where BFS would be more appropriate than DFS, or vice versa.

Discussion Prompt

Pose the question: 'Imagine you are designing a game where players explore a dungeon. Which traversal algorithm, BFS or DFS, would be better suited for finding all hidden treasures, and why? Be ready to explain your reasoning using graph terminology.'

Frequently Asked Questions

What is a graph in computer science?

A graph is a data structure consisting of vertices (nodes) and edges (connections between vertices). Edges can be directed (one-way) or undirected, and weighted (with values) or unweighted. Graphs model many real-world structures including social networks, road maps, web page links, and dependency relationships between software components.

What is the difference between BFS and DFS in graph traversal?

Breadth-First Search visits nodes layer by layer, exploring all nodes at distance 1 before distance 2, guaranteeing the shortest path in unweighted graphs. Depth-First Search follows one branch as deep as possible before backtracking. BFS uses a queue to manage traversal order; DFS uses a stack, either explicitly or via the call stack in a recursive implementation.

What is BFS used for in real-world applications?

BFS is used wherever shortest paths in unweighted graphs matter: GPS navigation finding fewest turns, social network friend suggestions within k degrees, network packet routing, and puzzle-solving algorithms finding the minimum number of moves to reach a goal state. Its layer-by-layer structure guarantees the shortest path to each node it discovers.

How does active learning help students understand graph traversal algorithms?

Graph algorithms have a natural physical dimension: nodes and edges can be represented by students and connections between them. Simulations where students physically act out BFS and DFS following exact algorithm rules make the traversal order visible and contrast the two approaches directly, building structural intuition that helps students reason about more complex graph problems later.