Exercise 2: Deployment Order With a New Logging Service — Possible Solution
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GIVEN
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This chapter's own Step 4 graph: auth->api, database->api,
cache->api, api->frontend, api->notifications
New edge: api->logging (nothing depends on logging in return)

STEP 1: IN-DEGREES AT THE START
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auth:         0   (nothing points to it)
database:     0   (nothing points to it)
cache:        0   (nothing points to it)
api:          3   (incoming from auth, database, cache)
frontend:     1   (incoming from api)
notifications: 1  (incoming from api)
logging:      1   (incoming from api - the new edge)

STEP 2: RUN KAHN'S ALGORITHM
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Initial queue (in-degree 0): auth, cache, database

Process auth: output auth. No effect on any in-degree (auth's only
edge goes to api, which stays at in-degree 2 for now).
  Wait - auth->api exists, so processing auth decrements api's
  in-degree from 3 to 2.

Process cache: output cache. Decrements api's in-degree from 2
to 1.

Process database: output database. Decrements api's in-degree
from 1 to 0 - api joins the queue.

Process api: output api. Decrements frontend, notifications, and
logging each from 1 to 0 - all three join the queue.

Process frontend, notifications, logging (in that order): output
each in turn. No further edges to decrement.

RESULT
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Valid deployment order: auth, cache, database, api, frontend,
notifications, logging

logging slots in alongside frontend and notifications as one of
api's three dependents, all three becoming deployable at the exact
same moment - the instant api itself finishes deploying - since
none of the three have any dependency relationship with each other.

WHY THIS WORKS AS AN ANSWER
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Every service's in-degree is computed correctly including the new
logging service and api's own updated in-degree of 3, and Kahn's
algorithm is traced step by step exactly as this chapter's own
method specifies - a service only joins the queue the moment its
in-degree reaches zero, not before.