Q$3.$ The message passing network illustrated in Figure Q$3$ consists of five intermediate

nodes, and communication links between those nodes. A message can be successfully

passed between A and $B$ if there is a path between them along which all intermediate

nodes are functional. To meet service requirements, the network should be designed

such that the probability of a message being delivered from $A$ to $B$ exceeds $95\%.$

$($a$)$ i$.$ Use set notation to describe the vertex set and the edge set of the graph

illustrated in Figure Q$3.$

ii$.$ Explain what is meant by the phrase "degree of a vertex in a graph". Support

your answer by identifying the degree of the vertex labelled a in Figure Q$3.$

$($b$)$ If the probability that any intermediate node has failed is $0\mathrm{.}05,$ and the inter$-$

mediate nodes fail independently of each other, what is the probability that the

message gets delivered? Does this meet the specified service requirements?

$($c$)$ Intermediate node failure is dependent on ambient temperature, where the prob$-$

ability of failure is $0\mathrm{.}05$ if temperatures are below $25$ degrees Celsius, and $0\mathrm{.}25$ if

temperatures are at or above $25$ degrees Celsius. What is the maximum proba$-$

bility $p$ of temperature being above $25$ degrees Celsius that will allow us to meet

the service requirement? Explain your reasoning.