I usually draw it the other way round, but you're on the right track.
You can cable switch-switch-switch until you go dizzy OR you reach the connection
limit of your router (most home routers support a MAX of 10 concurrent connections).
You can also place a switch on EVERY port of the router and then switch-switch-switch again.
The control of your network is the ROUTER; a switch is a non-configurable device
and transparent to everything. 
Heck; for a roommate, I even have
modem--routerA--switch--routerB
HOWEVER, systems attached to A can not see/share with those on B which is
what we want.