Day 07Part 06- Examples based on Bipartite graph and Complete bipartite graph.

Day 07Part 06- Examples based on Bipartite graph and Complete bipartite graph.

A Bipartite Graph is a graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. A Complete Bipartite Graph is a special type of bipartite graph where every vertex from one set is connected to every vertex in the other set.

 Example 1: Checking If a Graph is Bipartite

Graph:

Consider a graph with vertices V = {A, B, C, D} and edges E = {(A, C), (A, D), (B, C), (B, D)}.

Solution:

1. Divide the vertices into two sets:

   • Set 1: {A, B}

   • Set 2: {C, D}

2. Each edge connects a vertex in Set 1 to a vertex in Set 2 (i.e., no two vertices in the same set are adjacent).

3. Since this condition is satisfied, the graph is bipartite.

 Example 2: Complete Bipartite Graph (K₃,₄)

A Complete Bipartite Graph Kₘ,ₙ has two sets of vertices with m and n vertices, where every vertex in one set is connected to all vertices in the other set.

Graph:

Consider K₃,₄, with two sets:

• Set X: {A, B, C} (3 vertices)

• Set Y: {D, E, F, G} (4 vertices)

Edges:

Each vertex from Set X connects to all vertices in Set Y:

• A → {D, E, F, G}

• B → {D, E, F, G}

• C → {D, E, F, G}

Since every vertex from one set is connected to all vertices of the other set, this is a Complete Bipartite Graph.

 Key Points:

A graph is bipartite if it can be colored using two colors without any adjacent nodes sharing the same color.
A complete bipartite graph Kₘ,ₙ connects every vertex in one set to every vertex in the other set.

Would you like more practice problems or algorithmic methods to check if a graph is bipartite?

Day 07Part 06- Examples based on Bipartite graph and Complete bipartite graph.

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