A bipartite graph is a type of graph in graph theory where the set of vertices can be divided into two disjoint sets, such that no two vertices within the same set are adjacent. In simpler terms, it means that the graph can be colored with two colors in such a way that vertices of the same color are never directly connected by an edge. This characteristic makes bipartite graphs particularly interesting and useful in various applications, including modeling relationships in bipartite networks, solving matching problems, and in algorithms for scheduling and partitioning.
Detection of Bipartite Graphs
Detecting whether a graph is bipartite involves checking if it is possible to divide the graph's vertices into two sets where edges only run between the sets and not within them. There are two common algorithms used to detect bipartite graphs: Depth-First Search (DFS) and Breadth-First Search (BFS).
Using Depth-First Search (DFS)
The DFS approach involves traversing the graph deeply from one vertex to another and coloring the vertices such that no two adjacent vertices share the same color. If at any point it is not possible to color a vertex differently from its neighbors, the graph is not bipartite.
Initialize a color array to store colors assigned to all vertices. Vertex number is used as the index in this array. The value '-1' indicates that no color is assigned to the vertex, '1' indicates the first color, and '0' indicates the second color.
Choose an arbitrary vertex and color it with the first color (1).
Perform DFS from the chosen vertex. For every visited vertex, color it with an alternate color than its parent.
If a vertex is required to be colored with the same color as its parent (indicating an odd-length cycle), the graph is not bipartite.
Using Breadth-First Search (BFS)
The BFS approach uses a similar concept but explores the graph level by level. This method is particularly useful for detecting if a connected graph is bip...
