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Not usually, no. If they are on the same switch then they are usually on the same broadcast domain and IP subnet and can send frames and packets directly between them without the router.
In a home or small office setup the router is usually the DHCP server so needs to be working to allocate IP addresses but if you statically configured the IP addresses you wouldn't even need the router (although lack of DNS would confuse many).
jellybaby
@Polygon
Can someone clear something up for me? If two computers are connected to the same switch they'll communicate with each other via the network's router unless the switch is a layer 3 one. Is that correct?