Identifying vertices of a planar diagram

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In summary, a planar diagram is a geometric representation of points and lines on a flat surface. Vertices in a planar diagram are points where two or more lines intersect, represented by small circles or dots. To identify vertices, one can look for intersections or count corners in a shape. It is important to identify vertices in a planar diagram as they help determine its structure and relationships. A planar diagram can have multiple sets of vertices, depending on the context and purpose of the diagram.
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ascky
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Hi all,

I'm having trouble understanding what 'identifying vertices' means for a planar model of a surface. This is in the context of trying to classify a closed, connected surface to see whether it is homeomorphic to the sphere, or connected sums of tori, or connected sums of projective planes.

My text says something like 'we have two adjacent vertices in different equivalence classes'. How do I see which equivalence class the vertex belongs to? I thought only the edges of the diagram belonged to equivalence classes.

Thanks in advance.
 
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Identifying vertices means to assign a specific vertex in the planar model to a particular equivalence class. This is done by looking at the edges that connect the vertex with other vertices. If two or more edges connect the vertex to different equivalence classes, then the vertex is identified with one of the equivalence classes. This allows us to determine which equivalence class the vertex belongs to.
 

1. What is a planar diagram?

A planar diagram is a geometric representation of a set of points and lines on a flat surface, such as a piece of paper or a computer screen. It is also known as a planar graph or a planar embedding.

2. What are vertices in a planar diagram?

Vertices are the points in a planar diagram where two or more lines intersect. They are represented by small circles or dots in the diagram.

3. How do you identify vertices in a planar diagram?

To identify vertices in a planar diagram, look for points where two or more lines intersect. These points will be represented by small circles or dots. You can also count the number of corners in a shape, as each corner represents a vertex.

4. Why is it important to identify vertices in a planar diagram?

Identifying vertices in a planar diagram is important because they are crucial in determining the structure and properties of the diagram. Vertices can help in understanding the connectivity and relationships between different elements in the diagram.

5. Can a planar diagram have more than one set of vertices?

Yes, a planar diagram can have multiple sets of vertices. For example, a cube has 8 vertices, but if you break it down into 6 squares, each square has its own set of 4 vertices. It is important to consider the context and purpose of the diagram when identifying and labeling vertices.

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