Understanding Planar Graphs

What is a planar graph?

• A graph that can be drawn on a plane without edges crossing.

• A graph that can only be drawn in three dimensions.

• A graph with at least one edge crossing.

• A graph that cannot be represented on a plane.

Which of the following graphs is planar: K5, K3,3, or a triangle?

• square

• K5

• triangle

• K3,3

Can a graph with 5 vertices and 10 edges be planar?

• Yes, if the vertices are connected in a specific way

• It depends on the arrangement of edges

• No

• Yes, it can be planar

What is the maximum number of edges in a planar graph with n vertices?

• n^2 - n

• 3n - 6

• 2n - 4

• n + 1

Is a complete graph with 4 vertices planar?

• Yes, a complete graph with 4 vertices is planar.

• A complete graph with 4 vertices has more than 4 edges, making it non-planar.

• A complete graph with 4 vertices can be drawn without edges crossing.

• No, a complete graph with 4 vertices is not planar.

How can you determine if a graph is planar using Kuratowski's theorem?

• A graph is planar if it does not contain a subgraph that is a subdivision of K5 or K3,3.

• A graph is planar if it contains a subgraph that is a subdivision of K4.

• A graph is planar if it can be drawn without any edges crossing.

• A graph is planar if it has at least three vertices.

What is the significance of the Euler's formula in planar graphs?

• Euler's formula establishes a relationship between vertices, edges, and faces in planar graphs.

• It describes the color of the graph's edges.

• It calculates the shortest path between two vertices.

• It provides a method for coloring the vertices of a graph.

Provide an example of a real-world application of planar graphs.

• Analyzing weather patterns for climate studies.

• Designing electrical circuits to optimize space and avoid overlaps.

• Mapping social networks for user interactions.

• Creating 3D models for architectural designs.

What is the relationship between planar graphs and graph coloring?

• Graph coloring is only applicable to non-planar graphs.

• All graphs can be colored with only two colors regardless of their structure.

• Planar graphs require at least five colors for proper coloring.

• Planar graphs can be colored with at most four colors without adjacent vertices sharing the same color.

Can a graph be non-planar if it contains a subgraph that is planar?

• A graph cannot be non-planar if it has any planar subgraphs.

• All subgraphs of a non-planar graph must also be non-planar.

• Yes, a graph can be non-planar if it contains a subgraph that is planar.

• A graph is always planar if it contains a planar subgraph.