Question

(a) Determine the rook polynomial of the given board of shaded squares. (You may useany method of your choice.) (b) Radio stations broadcast their signal at certain frequencies. There are a limitednumber of frequencies to choose from, so many stations use the same frequency aslong as they are positioned far enough apart not to cause interference. The following8 stations A,..., H wish to broadcast in a particular region. The radio stations thatare close enough to each other to cause interference are given in the table below. (i) Represent this information as a graph G with vertices as the stations and edges when two stations are close enough to cause interference. (ii) Give with reasons, non-trivial upper and lower bounds on x(G) and find a minimum vertex colouring. Hence determine the minimum number of frequencies that are required. Write down a vertex decomposition of the graph to assign the radio stations to the frequencies.(5 marka) (iii) Suppose a new radio station, I, wishes to broadcast. Due to location this cannot share a frequency with D, E or G. Is it possible for this new station to broadcast using the minimum number of frequencies determined in (ii). Explain, with full reasons, if this is not possible and give a vertex decomposition if it is possible.

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