One thing that I remember from the Electrical Engineering class in my undergradute study was a technique to quickly determine/locate/figure out the nodes of a circuit when doing node voltage analysis (NVA). This I remember so vividly because the result of my first quiz on NVA was really bad because the networks of the circuits were so complicated and so I incorrectly determined the nodes. During the discussion of the quiz, the lecturer came up to the blackboard and showed a really nice technique to quickly determine the nodes of the complicated circuits.

For example, consider the following circuit. How many nodes are there?

The nodes can be determined quickly by the following steps:
1. Put a highlighter in one of the wires as depicted below.

2. Highlight the whole wire without crossing over any component and without taking up the highlighter from the paper as depicted below. It is crucial that the highlighter tip must remain in touch with the paper during a single traversal so as to prevent any need for figuring out where you have incomplete traversal. Imagining the tip of the highlighter as a mouse and the wire to be highlighted as a tunnel will help. Every component will serve as dead ends. The goal is for the mouse to traverse the whole tunnel. Since the mouse cannot do teleportation, your highlighter tip must be in touch all the time until the whole tunnel is traversed.

3. Repeat Step 1 and Step 2 for the remaining unhighlighted wires. If no such wire remains, each highlighted section is a node as depicted below.

So, there are 2 nodes.