i-v graph, how to find current and its behavior
It's not completely clear what the task is, but I'll take a stab.  

If the problem says something like, "plot \$i\$ and \$v\$ for the following circuit," then what that essentially means is: pretend there's an element between the two nodes on the left of the circuit.  If that element has zero volts across it, calculate the current that flows through it.  Mark that data point on your i-v graph.  Now pretend that element has 1 Volt across it and calculate how much current flows through it; mark that data point.  Repeat this until you have a full characteristic curve.

Many times people will draw a "fictitious" ideal voltage source  at the two nodes to help them visualize where the "pretend" voltage comes from.