Dependency Graphs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Dependency Graphs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A dependency graph is a directed graph where nodes are variables and arrows show which variables directly influence which others.

Like a flowchart: A affects B, B affects C. Arrows show dependencies.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Dependency graphs reveal the causal or computational structure of a system — following the arrows shows which quantities must be computed or measured before others.

Common stuck point: A cycle in a dependency graph creates a circular dependency — neither variable can be determined without knowing the other, requiring simultaneous solution.

Sense of Study hint: Draw boxes for each variable and arrows showing 'A affects B.' Then follow the arrows to trace which variables influence which.

Worked Examples

Example 1

easy
Three tasks have the following dependencies: Task B depends on Task A, and Task C depends on Task B. Draw the dependency graph and determine a valid execution order.

Solution

  1. 1
    Identify the dependencies: A \to B (B depends on A) and B \to C (C depends on B).
  2. 2
    Draw the directed graph: A \to B \to C. An arrow from X to Y means X must be completed before Y.
  3. 3
    Perform a topological sort: start with the node that has no incoming edges (A), then B, then C.
  4. 4
    The valid execution order is A, B, C.

Answer

A \to B \to C
A dependency graph is a directed acyclic graph (DAG) where edges represent prerequisite relationships. A topological sort gives a valid ordering that respects all dependencies, ensuring no task is started before its prerequisites are complete.

Example 2

medium
Given the dependencies: D depends on A and B; E depends on B and C; F depends on D and E. Find all valid topological orderings.

Example 3

medium
Tasks A \to B, A \to C, B \to D, C \to D. What is the minimum number of stages to complete all tasks if independent tasks can run in parallel?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A project has tasks with dependencies: B depends on A; C depends on A; D depends on B and C. What is the minimum number of time steps needed if independent tasks can run in parallel?

Example 2

hard
Given the dependency graph with edges A \to C, A \to D, B \to D, B \to E, C \to F, D \to F, E \to F, determine whether adding the edge F \to A would create a problem, and explain why.
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Related Concepts

Functional Dependency

Background Knowledge

These ideas may be useful before you work through the harder examples.

functional dependency