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Bipartite Graph Checker

Bipartite Graph Checker


Bipartite Graph Checker: Understanding and Implementation

A bipartite graph is a special type of graph in which the set of vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. This property makes bipartite graphs useful in various applications, such as matching problems, network flow, and even in computer science algorithms.

In this article, we’ll discuss what a bipartite graph is, how to check if a graph is bipartite, and present a method to efficiently check bipartiteness using algorithms.

What is a Bipartite Graph?

A bipartite graph consists of two sets of vertices, say U and V, where every edge in the graph connects a vertex in set U to one in set V. No edge exists between two vertices within the same set. Formally, a graph is bipartite if the vertex set V can be split into two disjoint subsets U and V such that every edge connects a vertex in U to one in V.

Key Properties of Bipartite Graphs:

  1. Two sets of vertices: All edges must connect a vertex in set U to a vertex in set V.
  2. No edges within a set: There are no edges between vertices in U or between vertices in V.
  3. Graph colorability: A graph is bipartite if and only if it is 2-colorable. This means we can color the graph using only two colors such that no two adjacent vertices share the same color.

Checking if a Graph is Bipartite

The simplest way to determine if a graph is bipartite is by checking if we can color the graph using two colors such that no two adjacent vertices share the same color. This can be done using either Breadth-First Search (BFS) or Depth-First Search (DFS). Here, we will focus on the BFS approach.

BFS Algorithm for Bipartite Graph Checking

The BFS algorithm works by traversing the graph level by level. During traversal, each vertex is assigned one of two colors (often represented by 0 or 1). If, at any point during traversal, we find that two adjacent vertices have the same color, then the graph is not bipartite.

Steps to Check if a Graph is Bipartite Using BFS:

  1. Initialize colors: Create an array color[] to store the color of each vertex. Initialize all vertices with no color, usually represented by -1.
  2. Traverse each unvisited vertex: For each unvisited vertex, perform a BFS. Assign it a color (say, color 0).
  3. Assign colors during BFS: For every adjacent vertex, if it is uncolored, assign it the opposite color (color 1 if the current vertex is color 0, and color 0 if the current vertex is color 1). If the adjacent vertex is already colored and the color matches the current vertex, the graph is not bipartite.
  4. Repeat until all vertices are processed: If you complete the BFS without conflicts, then the graph is bipartite. If any conflict is found, the graph is not bipartite.

Pseudocode for BFS Bipartite Check:

pythonCopydef is_bipartite(graph):
    n = len(graph)
    color = [-1] * n  # -1 means uncolored

    def bfs(start):
        queue = [start]
        color[start] = 0  # Assign color 0 to the start vertex

        while queue:
            node = queue.pop(0)
            for neighbor in graph[node]:
                if color[neighbor] == -1:
                    # Assign opposite color to the neighbor
                    color[neighbor] = 1 - color[node]
                    queue.append(neighbor)
                elif color[neighbor] == color[node]:
                    # If neighbor has same color, not bipartite
                    return False
        return True

    # Check each connected component of the graph
    for i in range(n):
        if color[i] == -1:  # If not visited
            if not bfs(i):
                return False

    return True

Explanation of the Pseudocode:

  • The color[] array keeps track of the color assigned to each vertex. If a vertex has not been colored yet, it is marked as -1.
  • The BFS starts from an unvisited vertex and assigns it a color (0 in this case).
  • It then explores all adjacent vertices. If an adjacent vertex is not colored, it is assigned the opposite color of the current vertex.
  • If an adjacent vertex already has the same color as the current vertex, the graph cannot be bipartite, and the function returns False.
  • If all vertices are processed without conflicts, the graph is bipartite, and the function returns True.

Time Complexity

The time complexity of this algorithm is O(V + E), where V is the number of vertices and E is the number of edges. This is because, in the worst case, we visit each vertex once and each edge once during the BFS traversal.

Applications of Bipartite Graphs

Bipartite graphs have numerous applications in computer science and mathematics, including:

  1. Matching problems: Bipartite graphs are widely used in matching algorithms, such as job assignments, dating algorithms, and network flow problems.
  2. Graph coloring: Since bipartite graphs are 2-colorable, they are used in problems that require coloring a graph with two colors.
  3. Network flow: Bipartite graphs are often used in algorithms for maximum flow and minimum cut problems.
  4. Social networks: Bipartite graphs model relationships between two distinct types of entities, like users and products or students and courses.

Conclusion

Checking if a graph is bipartite is an important problem in graph theory with applications in network flow, matching problems, and more. Using algorithms like BFS or DFS, we can efficiently determine whether a graph is bipartite. The BFS-based approach provides an elegant and scalable solution to this problem, making it applicable to large graphs with many vertices and edges.

By understanding the concept of bipartite graphs and how to check them, you open doors to solving a variety of graph-related problems in computational theory and real-world applications.

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