How to delete unnecessary resistances in a resistance network

I am not sure if this completely answers your question; if not I hope it at least gets you further to an actual solution

Take for instance the second of the two networks you presented

g = Graph[{0 <-> 13, 13 <-> 14, 14 <-> 0, 0 <-> 1, 12 <-> 1, 0 <-> 12,
10 <-> 16, 4 <-> 9, 9 <-> 10, 10 <-> 4, 11 <-> 9, 11 <-> 10, 
1 <-> 2, 2 <-> 3, 1 <-> 4, 1 <-> 5, 1 <-> 6, 4 <-> 8, 8 <-> 7, 
5 <-> 7, 6 <-> 3, 7 <-> 3, 11 <-> 4, 12 <-> 17, 17 <-> 18, 
18 <-> 19, 4 <-> 15, 15 <-> 20, 20 <-> 4, 10 <-> 21}, 
VertexLabels -> "Name"]

with FindPath you can find all routes between two points in your network (1 and 11 in this case) as follows

sol = FindPath[g, 1, 11, Infinity, All]

A simple visualization for this could be

Manipulate[HighlightGraph[g, PathGraph[sol[[n]]]], {n, 1, Length@sol, 1}]

To get the network containing only the relevant nodes you could use Subgraph as in

Subgraph[g, sol // Flatten // DeleteDuplicates, VertexLabels -> "Name"]