Bipartite Graph Checking - CSU083 | Shoolini University

Bipartite Graph Checking

2025, February 6

1. Prerequisites

Before understanding bipartite graph checking, you need to be familiar with the following concepts:

2. What is a Bipartite Graph?

A bipartite graph is a graph where its vertices can be divided into two disjoint sets, U and V, such that no two vertices within the same set are adjacent.

Formally, a graph G = (V, E) is bipartite if there exists a partition (U, V) where every edge (u, v) ∈ E satisfies u ∈ U and v ∈ V.

Key properties:

3. Why Does Bipartite Graph Checking Exist?

Bipartite graph checking is essential because bipartite graphs have unique properties that make them useful in:

4. When Should You Use Bipartite Graph Checking?

You should check whether a graph is bipartite in the following scenarios:

5. How Does Bipartite Graph Checking Compare to Alternatives?

Bipartite graph checking has its strengths and weaknesses when compared to other graph analysis methods:

Strengths:

Weaknesses:

6. Basic Implementation

The following is a Python implementation of bipartite graph checking using Breadth-First Search (BFS):


from collections import deque

def is_bipartite(graph, start):
    color = {}  # Dictionary to store colors (0 or 1)
    queue = deque([start])
    color[start] = 0  # Assign first color

    while queue:
        node = queue.popleft()
        for neighbor in graph[node]:
            if neighbor not in color:
                color[neighbor] = 1 - color[node]  # Assign alternate color
                queue.append(neighbor)
            elif color[neighbor] == color[node]:
                return False  # Same color adjacent nodes → Not bipartite
    return True

# Example Graph (Adjacency List Representation)
graph = {
    0: [1, 3],
    1: [0, 2],
    2: [1, 3],
    3: [0, 2]
}

# Checking bipartiteness for all components
print(is_bipartite(graph, 0))  # Output: True

Explanation:

7. Dry Run of the Algorithm

Let's dry run the algorithm on the following input graph:

  0 --- 1
  |     |
  3 --- 2

Step-by-Step Execution:

Step Queue Current Node Neighbor Color Map Action
1 [0] - - {0: 0} Start with node 0, color it 0
2 [] 0 1 {0: 0, 1: 1} Color 1 as 1, add to queue
3 [1] 0 3 {0: 0, 1: 1, 3: 1} Color 3 as 1, add to queue
4 [1, 3] 1 2 {0: 0, 1: 1, 3: 1, 2: 0} Color 2 as 0, add to queue
5 [3, 2] 3 2 {0: 0, 1: 1, 3: 1, 2: 0} 2 is already colored correctly (no conflict)
6 [2] 2 1 {0: 0, 1: 1, 3: 1, 2: 0} 1 is already colored correctly (no conflict)
7 [] - - {0: 0, 1: 1, 3: 1, 2: 0} Queue empty, return True

Final Output: The graph is bipartite.

Key Observations:

  • The algorithm colors nodes as we traverse and ensures adjacent nodes have different colors.
  • If at any point a node and its neighbor have the same color, the graph is not bipartite.
  • The BFS ensures that all components of the graph are checked.

8. Time & Space Complexity Analysis

8.1 Worst-Case Complexity (O(V + E))

In the worst case, the graph is fully connected and the algorithm must traverse all nodes and edges:

Total Complexity: O(V + E) (Linear Time Complexity)

8.2 Best-Case Complexity (O(1))

If the graph is empty (contains no edges), we return True immediately in O(1) time.

8.3 Average-Case Complexity (O(V + E))

For most graphs, the algorithm performs a BFS traversal, leading to an average complexity of O(V + E).

9. Space Complexity Analysis

Space consumption depends on the graph representation and auxiliary structures:

Total Space Complexity: O(V + E)

10. Trade-Offs Analysis

10.1 Strengths

10.2 Weaknesses

10.3 Alternative Approaches

11. Optimizations & Variants

11.1 Common Optimizations

11.2 Variants of Bipartite Graph Checking

12. Iterative vs. Recursive Implementations

12.1 Iterative (BFS) Approach

Using an explicit queue for traversal ensures O(V + E) time complexity without additional recursion overhead.


from collections import deque

def is_bipartite_bfs(graph, start):
    color = {}  
    queue = deque([start])
    color[start] = 0  

    while queue:
        node = queue.popleft()
        for neighbor in graph[node]:
            if neighbor not in color:
                color[neighbor] = 1 - color[node]  
                queue.append(neighbor)
            elif color[neighbor] == color[node]:
                return False  
    return True

Pros of Iterative BFS:

12.2 Recursive (DFS) Approach

The recursive DFS approach avoids using an explicit queue but may run into recursion depth issues on large graphs.


def is_bipartite_dfs(graph, node, color, current_color):
    if node in color:
        return color[node] == current_color  

    color[node] = current_color  

    for neighbor in graph[node]:
        if not is_bipartite_dfs(graph, neighbor, color, 1 - current_color):
            return False  

    return True

def check_bipartite(graph):
    color = {}
    for node in graph:
        if node not in color:
            if not is_bipartite_dfs(graph, node, color, 0):
                return False
    return True

Pros of Recursive DFS:

Cons of Recursive DFS:

12.3 Summary of Efficiency Comparison

Method Time Complexity Space Complexity Best Use Case
Iterative BFS O(V + E) O(V) Large graphs, avoiding recursion depth issues
Recursive DFS O(V + E) O(V) (function call stack) Smaller graphs, cleaner implementation

Final Takeaway: Use BFS for large graphs to avoid recursion depth issues. DFS is more readable but less scalable.

13. Edge Cases & Failure Handling

When checking for bipartiteness, certain edge cases and pitfalls can cause incorrect results or inefficiencies.

13.1 Common Edge Cases

13.2 Handling Failures

14. Test Cases to Verify Correctness

Below are Python test cases to verify the correctness of the bipartite checking algorithm.


def test_bipartite():
    test_cases = [
        {
            "graph": {0: [1], 1: [0]},  # Simple bipartite graph
            "expected": True
        },
        {
            "graph": {0: [1, 2], 1: [0, 3], 2: [0, 3], 3: [1, 2]},  # Square (even cycle)
            "expected": True
        },
        {
            "graph": {0: [1, 2], 1: [0, 2], 2: [0, 1]},  # Triangle (odd cycle)
            "expected": False
        },
        {
            "graph": {0: []},  # Single node
            "expected": True
        },
        {
            "graph": {0: [0]},  # Self-loop
            "expected": False
        },
        {
            "graph": {0: [1], 1: [2], 2: [3], 3: [0], 4: [5], 5: [4]},  # Two components, both bipartite
            "expected": True
        },
        {
            "graph": {0: [1, 2], 1: [2, 3], 2: [0, 1, 3], 3: [1, 2]},  # Odd-length cycle in a component
            "expected": False
        }
    ]

    for i, test in enumerate(test_cases):
        result = check_bipartite(test["graph"])
        assert result == test["expected"], f"Test case {i} failed: expected {test['expected']}, got {result}"
    
    print("All test cases passed!")

test_bipartite()

14.1 Test Coverage

15. Real-World Failure Scenarios

15.1 Network Partitioning Errors

In distributed networks, incorrect bipartite checking can cause partitioning failures in network routing, leading to performance issues.

15.2 Scheduling Conflicts

Incorrectly assuming a bipartite structure in scheduling applications can lead to infeasible assignments (e.g., a conflict in job scheduling).

15.3 Social Network Analysis Errors

Many recommendation systems assume bipartite graphs for collaborative filtering. Misclassification can degrade recommendation quality.

15.4 Incorrect Graph Processing in Machine Learning

In graph-based ML, assuming bipartiteness can introduce data biases if the graph is actually non-bipartite.

16. Real-World Applications & Industry Use Cases

16.1 Job Matching & Assignment Problems

Bipartite graphs are widely used in matching problems where one set of entities must be paired with another.

16.2 Network Flow & Optimization

Many network problems can be modeled as bipartite graphs to optimize flow and routing.

16.3 Recommendation Systems

Recommendation systems use bipartite graphs to model interactions between users and items.

16.4 Security & Fraud Detection

Bipartite graphs can help detect fraud and anomalies in networks.

16.5 Biological & Medical Research

Bipartite graphs help analyze relationships in biological networks.

17. Open-Source Implementations

17.1 NetworkX (Python)

NetworkX is_bipartite() provides a built-in way to check if a graph is bipartite.


import networkx as nx

G = nx.Graph()
G.add_edges_from([(0, 1), (1, 2), (2, 3), (3, 0)])
print(nx.is_bipartite(G))  # Output: True

17.2 Boost Graph Library (C++)

Boost Graph Library (BGL) provides efficient graph processing functions.

17.3 igraph (R, Python, C)

igraph is a powerful tool for handling large-scale graphs.

18. Practical Project: Matching Freelancers to Projects

18.1 Project Overview

We will implement a real-world script that assigns freelancers to projects using bipartite graph matching.

18.2 Problem Statement

Given a list of freelancers with different skills and a list of projects requiring specific skills, we must determine if a perfect matching exists.

18.3 Implementation


from collections import deque

def is_bipartite(graph):
    color = {}
    queue = deque()

    for start in graph:
        if start not in color:
            queue.append(start)
            color[start] = 0  

            while queue:
                node = queue.popleft()
                for neighbor in graph[node]:
                    if neighbor not in color:
                        color[neighbor] = 1 - color[node]
                        queue.append(neighbor)
                    elif color[neighbor] == color[node]:
                        return False
    return True

# Example Data
freelancers = {"Alice": ["Python", "JavaScript"], "Bob": ["Java", "C++"], "Charlie": ["Python", "C++"]}
projects = {"Project1": ["Python"], "Project2": ["Java"], "Project3": ["C++"]}

# Creating a Bipartite Graph
graph = {}
for freelancer, skills in freelancers.items():
    graph[freelancer] = []
    for project, required_skills in projects.items():
        if any(skill in skills for skill in required_skills):
            graph[freelancer].append(project)
            graph.setdefault(project, []).append(freelancer)

# Checking if the Assignment is Possible
print("Is bipartite (possible assignment)?", is_bipartite(graph))

18.4 How This Works:

18.5 Further Enhancements:

19. Competitive Programming & System Design Integration

19.1 Competitive Programming Use Cases

Bipartite graph checking frequently appears in contests due to its applications in:

19.2 Competitive Programming Strategies

19.3 System Design Integration

Bipartite graphs play a key role in designing scalable applications:

Example: In a microservices architecture, bipartite graphs can be used to model service dependencies, ensuring that no cyclic dependencies exist in a dependency graph.

20. Assignments

20.1 Solve 10 Competitive Programming Problems

Practice solving at least 10 problems on platforms like Codeforces, LeetCode, AtCoder, or GeeksforGeeks:

  1. Check if a graph is bipartite – Basic implementation (LeetCode Medium).
  2. Maximum Bipartite Matching – Hopcroft-Karp Algorithm (Codeforces).
  3. Detect Odd-Length Cycle – Use BFS or DFS (AtCoder).
  4. Graph Coloring – Check if a graph is 2-colorable (GeeksforGeeks).
  5. Stable Matching Problem – Gale-Shapley Algorithm (ICPC Archives).
  6. Minimum Vertex Cover – Reduce to bipartite matching (Hackerrank).
  7. Graph Theory Problems in Networks – Apply bipartite properties to network routing (LeetCode).
  8. Friend Recommendations – Use bipartite models for user-item recommendations (Codeforces).
  9. Course Scheduling – Ensuring dependencies can be resolved (TopCoder).
  10. Two-Sets Problem – Splitting elements into two valid groups (GeeksforGeeks).

20.2 Use Bipartite Graphs in a System Design Problem

Design a real-world application where bipartite graphs help solve a critical problem.

Example: Ride-Sharing System

20.3 Implement Under Time Constraints

Time yourself and try solving a problem within strict time limits:

Practicing under constraints improves problem-solving speed for coding competitions.