Question B$6$:

A city is trying to decide between two potential designs for their water network.

Each vertex is an access point, and each edge is a segment of pipe. In the future, they will occassionally need to open an access point and send a robot through the network to inspect pipes looking for leaks. After the robot is done, it can be removed from either the same access point, or a different access point.

$($a$)$ Which network should they choose if it is important that the robot inspect every section of pipe exactly once? $($The inspection would involve an Euler path or Euler cycle.$)$

$($b$)$ Label the vertices of the graph, and describe an Euler path by listing the vertices in the order the robot visits them. $($A pdf of each unlabelled graph is provided as a starting point, or you may draw the graph you selected by hand$).$

$($c$)$ Why is the other network not suitable for this purpose?