Question

n sensors all having range equal to 2, form a unit line graph arranged on a line such that
the ith sensor has x-coordinate equal to xi = i - 1/i, for i = 1; 2; ....... n.
1. [1 pt] List by coordinate the first 5 vertices (left to right) of the resulting graph.
2. [1 pt] List by coordinate pairs the first 5 edges (left to right) of the resulting graph.
3. [1 pts] Is the resulting graph connected and why?

Answer 1

1). The x-coordinates of the first five vertices(left to right) of the resulting graph are :-

= = 0

= =  

=   =

=   =

= =  

2). The coordinate pairs of the first 5 edges of the resulting graph are as follows :-

=

=

=  

=

=

3) Yes, the resulting graph is connected . This is so because every vertex is connected to it's next adjacent vertex .

This is so because distance between two adjacent vertices and is given by - = =   = for   

Now, has the maximum value at which is 0.5

Thus, distance between two consecutive vertices - =     1.5

Since each sensor has a range equal to 2 and distance between two consecutive vertices is  at most 1.5 , two adjacent vertices will always be connected . Thus, we have a path between any two vertices in the above line graph.Thus, we can see that the resulting graph is connected.