Planar Graph Explanation:

Planar graph is a type of graph in which no two branches cross each other. This property makes planar graphs particularly interesting and useful in various applications, such as circuit design, network analysis, and transportation planning.

Definition of Planar Graph:

A graph is called planar if it can be drawn on a plane in such a way that no two edges intersect except at their endpoints. In other words, the graph can be represented without any edge crossings.

Characteristics of Planar Graph:

1. No edge crossings: In a planar graph, no two edges cross each other, which means the edges do not intersect except at the vertices.
2. Embedded on a plane: The graph can be drawn on a flat surface (plane) without any edge crossings.
3. Regions: The plane divides into regions by the edges and vertices of the graph, forming a unique structure.

Importance of Planar Graphs:

1. Simplified visualization: Planar graphs offer a clear and easy-to-understand representation due to the absence of edge crossings.
2. Efficient algorithms: Many algorithms are designed specifically for planar graphs, taking advantage of their unique properties to achieve faster and more efficient computations.
3. Practical applications: Planar graphs are commonly used in various real-world scenarios, such as designing electronic circuits, mapping transportation networks, and analyzing social connections.

In conclusion, a graph is called a planar graph if its branches do not cross each other when represented on a plane. This property makes planar graphs valuable in different fields of study and practical applications.