The Floo Network

Harry is going to be late to his class, so he needs to move as quickly as possible.

Here is a map of his institution, where the edges represent roads he could walk through, and the nodes of the same color are those between which he could teleport using the Floo Network:

What is the least number of roads he would need to walk through to get to the classroom?

Clarification: Teleportations happen instantaneously and thus are not included in the count.

Inspired by Christopher Boo


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