Tools

Cycles in a graph

Objective

This interactive experiment introduces you to fundamental concepts in graph theory through engaging, hands-on exploration. You will master three important types of cycles in graphs while developing problem-solving skills that are essential in computer science, mathematics, and real-world applications.

Learning Goals

By the end of this experiment, you will be able to:

  • Identify and construct Hamiltonian cycles - paths that visit every vertex exactly once
  • Recognize and validate Eulerian cycles - paths that traverse every edge exactly once
  • Solve the Traveling Salesman Problem (TSP) - finding the shortest route through all vertices
  • Understand the theoretical foundations behind these classical graph problems

Getting Started

Interface Overview

The simulation provides an intuitive visual interface with several key components:

  1. Interactive Graph Canvas: A circular arrangement of labeled vertices (A, B, C, etc.) connected by edges
  2. Challenge Selector: Dropdown menu to choose between Hamiltonian, Eulerian, and TSP challenges
  3. Control Buttons: Tools for managing your exploration (New Graph, Clear Path, Undo, Check Cycle)
  4. Floating Panels:
    • Graph Parameters Panel (gear icon): Adjust vertex count (4-8) and edge density (0.3-0.9)
    • Information Panel (info icon): Quick reference for cycle types and their properties
  5. Feedback System: Real-time guidance and validation of your solutions

Basic Interaction

  • Click vertices to build your path step by step
  • Follow the visual feedback - selected vertices turn red, path vertices turn green
  • Monitor your progress in the current path display
  • Use undo to backtrack if you make a mistake

Experiment Procedures

Task 1: Mastering Hamiltonian Cycles

Objective: Learn to identify and construct paths that visit every vertex exactly once.

Step 1: Setup and Understanding

  1. Select the challenge: Choose "Find Hamiltonian Cycle" from the dropdown menu
  2. Read the prompt: Notice the challenge description updates to guide you
  3. Observe the graph: Study the vertex arrangement (A, B, C, D, E, F) and edge connections
  4. Check the information panel: Review Hamiltonian cycle properties if needed

Step 2: Strategy Development

  1. Count the vertices: Note how many vertices need to be visited (typically 6)
  2. Identify potential starting points: Any vertex can serve as a starting point
  3. Look for dead ends: Identify vertices with only one or two connections
  4. Plan your route: Consider which path might allow you to visit all vertices

Step 3: Interactive Construction

  1. Start your cycle: Click any vertex to begin (it will turn red)
  2. Build your path:
    • Click adjacent vertices to extend your path
    • Watch as selected vertices turn green
    • The current path displays at the top
  3. Handle feedback:
    • Red error messages indicate invalid moves (no edge exists, vertex already visited)
    • Continue building until you can return to your starting vertex
  4. Complete the cycle: Click your starting vertex again to close the cycle

Step 4: Validation and Learning

  1. Automatic checking: The system validates your cycle automatically upon completion
  2. Interpret results:
    • ✅ Success: "Valid Hamiltonian cycle!" - you've found a correct solution
    • ❌ Error: "Invalid Hamiltonian cycle" - review what went wrong
  3. Try the undo feature: Practice backtracking using the Undo button
  4. Experiment further: Use "Clear Path" to try different approaches

Step 5: Advanced Exploration

  1. Generate new graphs: Click "New Graph" to practice with different structures
  2. Adjust difficulty: Use the parameters panel to:
    • Increase vertex count (more challenging)
    • Adjust edge density (affects solution availability)
  3. Challenge yourself: Try to find Hamiltonian cycles in graphs with fewer edges

Task 2: Exploring Eulerian Cycles

Objective: Understand paths that traverse every edge exactly once.

Step 1: Theory Application

  1. Select Eulerian mode: Choose "Find Eulerian Cycle" from the dropdown
  2. Check vertex degrees: Open the Graph Parameters panel to view degree sequence
  3. Apply Euler's theorem:
    • An Eulerian cycle exists only if all vertices have even degree
    • If any vertex has odd degree, no Eulerian cycle is possible

Step 2: Feasibility Analysis

  1. Examine the feedback: The system will indicate if an Eulerian cycle is possible
  2. If possible: Proceed to construct the cycle
  3. If impossible: The system explains why (odd degree vertices exist)
  4. Generate new graphs until you find one with all even-degree vertices

Step 3: Cycle Construction (for valid graphs)

  1. Start at any vertex: Unlike Hamiltonian cycles, starting point doesn't matter
  2. Traverse edges systematically:
    • Click vertices to move along edges
    • Key difference: You may revisit vertices, but never reuse edges
    • Track your progress - each edge should be used exactly once
  3. Plan carefully: Avoid getting trapped with no way to return to start

Step 4: Validation Process

  1. Complete the cycle: Return to your starting vertex
  2. System validation: Checks that every edge was used exactly once
  3. Learn from mistakes: If invalid, consider which edges were missed or repeated

Task 3: Solving the Traveling Salesman Problem

Objective: Find the shortest Hamiltonian cycle by minimizing total edge weights.

Step 1: Understanding Weighted Graphs

  1. Select TSP mode: Choose "Find TSP Cycle" from the dropdown
  2. Observe edge weights: White squares on edges show numerical weights (1-9)
  3. Understand the goal: Find a Hamiltonian cycle with minimum total weight

Step 2: Strategic Planning

  1. Identify heavy edges: Note which edges have high weights to avoid
  2. Look for light edges: Prioritize paths using low-weight edges
  3. Consider trade-offs: Sometimes a longer path with lighter edges is better

Step 3: Solution Construction

  1. Build a Hamiltonian cycle: First ensure you visit all vertices exactly once
  2. Calculate as you go: Mental math helps - add edge weights in your path
  3. Compare alternatives: Try different routes to find the optimal solution

Step 4: Optimization and Validation

  1. Submit your solution: Complete the cycle and let the system check it
  2. Interpret feedback:
    • Optimal: "Optimal TSP cycle found! Total weight: X"
    • Suboptimal: "Your path weight is X, but a better path with weight Y exists"
  3. Iterative improvement: Use the feedback to find better solutions

Advanced Features and Tips

Utilizing the Auto-Complete Feature

  • Automatic assistance: After 30 seconds of inactivity, the system provides solutions
  • Learning opportunity: Compare your approach with the automated solution
  • Don't rely on it: Try to solve problems independently first

Parameter Experimentation

  1. Vertex count effects:

    • 4 vertices: Simple, good for learning basics
    • 6 vertices: Standard difficulty, balanced complexity
    • 8 vertices: Advanced, requires careful planning

  2. Edge density impact:

    • Low density (0.3): Fewer solutions, more challenging
    • High density (0.9): Multiple solutions, easier to find cycles

Problem-Solving Strategies

For Hamiltonian Cycles

  • Start with vertices that have few connections
  • Avoid creating "islands" of unvisited vertices
  • Use systematic exploration rather than random clicking

For Eulerian Cycles

  • Check degree parity first - don't waste time on impossible graphs
  • Follow a methodical path to avoid missing edges
  • Remember: vertex revisiting is allowed, edge reuse is not

For TSP

  • Use a greedy approach initially (choose lightest available edges)
  • Consider the "nearest neighbor" heuristic
  • Remember that the globally optimal solution may require some heavier local choices