In this article, we will explore how to use DFS to check if a graph is bipartite. Before getting into the details, let us have a brief refresher on the concept of a bipartite graph. A bipartite graph is a graph which has its nodes divided into two independent sets, U and V. Every edge in the graph connects a node from U to a node from V.

What is a Bipartite Graph?

A bipartite graph is a graph G(U,V,E), where U and V are independent sets of vertices and E is the set of edges. Every edge in E connects a vertex in U to a vertex in V. A graph is said to be bipartite if its vertices can be divided into two disjoint sets such that all its edges run between the two sets.

What is Depth-First Search (DFS)?

Depth-First Search (DFS) is an algorithm used to traverse the vertices of a graph. It starts at the root of the tree and explores each branch before exploring the next. It checks all the nodes at the deepest level first before going one level up and continues in a similar fashion until it reaches the source node.

How to Use DFS to Check if a Graph is Bipartite?

Depth-First Search (DFS) can be used to check if a graph is bipartite. To do so, we follow the steps given below:

1. Initialize all the nodes in the graph to "unvisited".
2. Color the first node encountered as “red”.
3. Do a DFS traversal of the graph.
4. For each node encountered in the graph, color it with the opposite color of its predecessor.
5. If we encounter a node that has the same color as its predecessor, the graph is not bipartite.

FAQs:

Q1. What is a bipartite graph?

A bipartite graph is a graph which has its nodes divided into two independent sets, U and V. Every edge in the graph connects a node from U to a node from V.

Q2. How are nodes colored in a bipartite graph?

Nodes in a bipartite graph can be colored in two different colors, 'red' and 'blue'.

Q3. What is depth-first search?

Depth-first search (DFS) is an algorithm used to traverse the vertices of a graph. It starts at the root of the tree and explores each branch before exploring the next.

Q4. How can DFS be used to check if a graph is bipartite?

DFS can be used to check if a graph is bipartite. To do so, we have to first initialize all nodes in the graph to "unvisited" and color the first node encountered as “red”. Then, do a DFS traversal of the graph. For each node encountered in the graph, color it with the opposite color of its predecessor. If we encounter a node that has the same color as its predecessor, the graph is not bipartite.

Q5. What is the time complexity of DFS?

The time complexity of DFS for a graph G with V vertices and E edges is O(V+E).