Q4. - Geometry

Question

The number of odd vertices will tell you whether or not a network can be   traced without backtracking. Do you see how? If not, read on.

suppose that a given network can be traced without backtracking.

     a. Consider a vertex that is neither the start nor end of a journey through this network. Is such a vertex odd or even?

     b. Now consider the two vertices at the start and finish of a journey through this network. Can both of these vertices be odd? Even?

     c. Can just one of the start and finish vertices be odd?  

Step-by-Step Solution

Answer

a. The  vertex is even.

b. The  answer is yes, the vertices can be  even or odd.

c. The final answer is no.

Part a. Step 1. Given information:

The  given figure is as follows:

Step 2. Concept use.

A vertex with odd number of edges attached to it is called odd vertex and a vertex with even number of edges attached to it is called even vertex.

Step 3. Applying the concept.

A vertex will be even because as it is given in the problem that the vertex is not start or end of the journey via network then it is necessary pass it in another way, if it coming in one way.

  Hence, the vertex is even.

Part b. Step 1. Given information:

The  given figure is as follows:

Step 2. Concept use.

1. If vertices have same start and end points then both the vertices will be even.
2. If vertices have different start and end points then both the vertices will be odd.

Step 3. Applying the concept.

Both the vertices can be even or odd but it should be kept in mind either both the vertices will be even or both the vertices will be odd. Therefore, there can be two cases related to it. 

Case 1. If vertices have same start and end points then both the vertices will be even.

Case 2. If vertices have different start and end points then both the vertices will be odd.

 Hence, the vertex can be even or odd.

Part c. Step 1. Given information:

The  given figure is as follows:

Step 2. Concept use.

1. If vertices have same start and end points then both the vertices will be even.
2. If vertices have different start and end points then both the vertices will be odd.

Step 3. Applying the concept.

In  the given network, total number of vertices $=6.$

Only one of the start and finish vertices cannot be odd. This is because of the other vertex which is even. As that vertex is even, it is made to enter then it have to leave also. This makes it sure that other vertex should be neither start nor the end  point.

  Hence, only one of the start and finish vertices cannot be odd.