during installing router what networking components are needed in the internet protocol (tcp/ip) properties setup

What kind of router are you referring to?

On a top/ip network, for the hosts to communicate the need to have a unique ip assign within the local subnet.

You need the ip and subnet mask at a bare minimum. To get out of the subnet, the hosts require a default gateway which is the ip of the router's interface.

For name resolution, you need DNS.

Please clarify your question.

IP routers are typically layer 3 devices, but certain models can route based on higher levels of the protocol stack.....

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