How can one find the network design from a single laptop connected through DHCP IP? No other privileges in hand.

Start with getting the local PC information using ipconfig /all or ifconfig to get the local IP address information. 

In windows, you will receive get back a significant amount of information.

First local IP address, subnet mask, and default gateway.

 Windows will also return the IP address of the DHCP server and DNS servers assigned to your computer.

You can use the ping and traceroute (tracert) commands to follow the routing path to the DNS and DHCP server.

The traceroute command will give you the name (sometimes) and IP address (sometimes) of the routing equipment between your computer and the DNS or DHCP servers. 

With these commands, you can start to find the network topology. 

This depends on the laptop's OS.  Below are examples:

1. Windows: run this commend from a command prompt: ipconfig /all
2. Linux, Solaris, MacOS, AIX: run ifconfig from a command prompt

Hope this helps ..

First we have to check wether ip routing is enabled or not

then check the ip route command.

then give the network topology command

with this we can know on which network we are working and how many neighbor we are connected with.