Question

What is Kurawtoski Graph?

Comments

i think you are talking about Kuratowski's theorem, that says “A graph is planar if and only if it does not have any subdivision of complete graph K5 or complete bipartite graph $K_{3,3}$”.   

Answer 1

The Kuratowski graph refers to a particular graph that is used to characterize planar graphs. In graph theory, a planar graph is a graph that can be drawn on a 2D plane without any of its edges crossing each other. In 1930, Kazimierz Kuratowski, a Polish mathematician, proved a fundamental theorem that establishes a criterion for determining whether a graph is planar or not.

Kuratowski's theorem states that a graph is non-planar if and only if it contains a subgraph that is a homeomorphic to either the complete graph on five vertices (denoted as K₅) or the complete bipartite graph on three vertices each (denoted as K₃,₃). Homeomorphic means that the subgraph can be obtained from K₅ or K₃,₃ by a series of edge contractions and edge deletions.

The Kuratowski graph itself is a simple graph with 11 vertices and 20 edges that serves as an example of a non-planar graph. It is formed by two copies of K₅ connected at a single vertex.

To summarize, the Kuratowski graph is an important concept in graph theory, used to demonstrate the non-planarity of graphs, and it plays a significant role in understanding the properties of planar graphs.