Why the resistor is not considered?
This is simply because the variable of interest is Vo. 

There is only one current running through a branch. The current through the 6 ohm is the same current running through the 4 ohm, which is the current through the branch. Since KCL deals with currents leaving or entering a node from its branches and the current through the 4 ohm is the same as the current through the branch, we don't have to include the 6 ohm resistor and the equation solves directly for V0.