Mail Delivery - Core idea

Core idea: This is a direct Euler circuit problem.

Conditions to check: $1.$ Every vertex has even degree

$2.$ The graph is connected (all vertices with edges are reachable from vertex $1$)

If either condition fails, output "IMPOSSIBLE".

Why must we start at vertex $1$? The problem says we start and end at the post office (vertex $1$). Since it's a circuit, we can start anywhere, but we need $1$ in our output.

If $1$ has no edges, output "IMPOSSIBLE" (can't leave the post office).