In resistor-network analysis, what fundamental criterion decides whether components are connected in series, in parallel, or in a series–parallel combination?

Introduction / Context:
Correctly classifying connections as series or parallel is essential for simplification and design. Students often look at the schematic drawing instead of the actual electrical relationships. The true classification depends on how current and voltage are shared among components and how nodes are connected.

Given Data / Assumptions:

• Passive resistive networks under DC or steady-state conditions.
• Ideal wires (no resistance) define nodes.
• No dependent sources inside the elements under discussion.

Concept / Approach:
Two components are in series if the same current must flow through both (they share a single intermediate node with no branching). Two components are in parallel if both terminals of one are connected to the corresponding two terminals of the other, so their voltages are identical while currents may differ. More complex networks combine these relations to form series–parallel structures.

Step-by-Step Solution:
Identify nodes: mark all points connected by ideal conductors.Check current path: if only one path exists through two elements sequentially, they are in series.Check voltage sharing: if elements share both end nodes, they are in parallel.If neither condition holds globally, the network is a series–parallel combination or a bridge network.

Verification / Alternative check:
Use Kirchhoff's laws: series elements share the same current (KCL), parallel elements share the same voltage (KVL around a small loop gives zero difference).

Why Other Options Are Wrong:

• Voltage/power source: The source type does not dictate connection; node topology and current paths do.
• Resistance value: Magnitude does not determine series/parallel status—connectivity does.

Common Pitfalls:

• Basing judgment on physical drawing proximity rather than node connectivity.
• Overlooking hidden branches (e.g., measurement instruments or loads) that break series conditions.

Final Answer:
current flow