$2.($Heuristic search; $100$ points$)$

A group of researchers are being chased by zombies through a dark forest late at night. They

come to a rope bridge over a ravine. If they can all get over the bridge before the zombies

arrive, they will survive.

At most two people can cross at a time. A person or pair of people can only cross when they

have a flashlight with them. The group has only a single flashlight among them, so one person

must bring the flashlight back across the bridge to the starting side before anyone else can

cross. Each pair moves at the pace of its slowest member; i$.$e$.,$ the Undergrad and the Professor

will take $10$ minutes to cross if they go together.

Each person moves at a different pace:

$$ The Undergrad can cross the bridge in $1$ minute

$$ The Grad Student can cross the bridge in $2$ minutes

$$ The Postdoc can cross the bridge in $5$ minutes

$$ The Professor can cross the bridge in $10$ minutes

How can the whole group get to the other side of the bridge in the shortest possible time?

$($a$)[20$ points$]$ Represent this problem as a search graph. Be sure to include and formally

describe each component of the search graph.