Is internal resistance of cell part of the equivalent resistance of the network of resistors? Do we include the internal resistance of cell while calculating equivalent resistance of network? Take, for instance, the question given. Do we include the 1 ohm internal resistance while calculating equivalent resistance of the network?
 A: From the wording "A network [...] to a battery with internal resistance" I would say the network is everything except the battery, where the battery consists of the ideal voltage source and the $1\,\Omega$ resistor. It is impossible to know for sure though what the author really meant.
A: Although the question is tagged "Homework and Exercise" I will answer it because it has been open for over a month with no votes to close, and since I think your question is a conceptual one.
The problem states "a network of resistors is connected to a 16 volt battery". The only way to "connect" the network of resistors to the battery is to make the connections to the battery terminals. The internal resistance of the battery is not "connected" to the battery terminals. It exists internal to the battery between its terminals. 
Since the problem asks for the equivalent resistance of the network of resistors and the network of resistors is connected to the battery terminals, the equivalent resistance should not include the battery internal resistance.
Note that if instead the problem asked for the equivalent resistance of the circuit shown, then that would include the internal resistance of the battery.
Hope this helps.
