4 The diagram shows a network of roads connecting 6 villages. The number on each edge is the length, in miles, of the road.\\
\includegraphics[max width=\textwidth, alt={}, center]{63e7775d-2a63-4584-b3be-ce97927bcfcc-04_670_1298_466_356}

Total length of the roads $= 164$ miles
\begin{enumerate}[label=(\alph*)]
\item A police patrol car based at village $A$ has to travel along each road at least once before returning to $A$. Find the length of an optimal 'Chinese postman' route for the police patrol car.
\item A council worker starts from $A$ and travels along each road at least once before finishing at $C$. Find the length of an optimal route for the council worker.
\item A politician is to travel along all the roads at least once. He can start his journey at any village and can finish his journey at any village.
\begin{enumerate}[label=(\roman*)]
\item Find the length of an optimal route for the politician.
\item State the vertices from which the politician could start in order to achieve this optimal route.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA D1 2006 Q4 [11]}}