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Transform a graph into the following one?

GROUPS:

Hello, Let us consider the graph2:

z3 = {1 <-> 2, 2 <-> 3, 3 <-> 4, 4 <-> 5, 5 <-> 6, 6 <-> 7, 7 <-> 19,19 <-> 20, 19 <-> 22,
20 <-> 21, 20 <-> 23, 7 <-> 8, 8 <-> 24, 24 <-> 25, 24 <-> 26, 8 <-> 9, 9 <-> 10, 9 <-> 27, 
27 <-> 28, 27 <-> 29, 10 <-> 11, 11 <-> 12, 12 <-> 13, 13 <-> 14, 14 <-> 15,15 <-> 16, 
16 <-> 17, 17 <-> 18, 12 <-> 30, 30 <-> 31, 31 <-> 32, 32 <-> 33, 30 <-> 34, 31 <-> 35, 
32 <-> 36, 34 <-> 37, 34 <-> 38, 2 <-> 39, 3 <-> 40, 4 <-> 41, 5 <-> 42, 6 <-> 43, 
10 <-> 44, 11 <-> 45, 13 <-> 46, 14 <-> 47, 15 <-> 48, 16 <-> 49, 17 <-> 50,
18 <-> 51, 18 <-> 52, 1 <-> 53, 1 <-> 54}; 
graph2= Graph[z3,GraphLayout -> "SpringEmbedding",(*VertexLabels\[Rule]"Name",*)EdgeStyle -> Thick]

The color version looks like this:

enter image description here

The question is: How can we get graph2 from the graph1?

z1 = {1 <-> 2, 1 <-> 3, 1 <-> 4, 1 <-> 5, 1 <-> 6, 1 <-> 7, 1 <-> 8,1 <-> 9, 1 <-> 10, 1 <-> 11, 
1 <-> 12, 9 <-> 13, 1 <-> 14, 1 <-> 15,1 <-> 16, 12 <-> 17, 1 <-> 18, 12 <-> 19, 12 <-> 20, 
1 <-> 21,1 <-> 22, 1 <-> 23, 8 <-> 24, 4 <-> 25, 4 <-> 26, 20 <-> 27}; 
graph1 = Graph[z1, GraphLayout -> "RadialDrawing"(*,VertexLabels\[Rule]"Name"*)]

enter image description here

Let us imagine that it is a network of streets. Red trail markes the longest street. The longest street connects directly to 18 smaller streets - this is the degree of the middle vertex in graph1. The second longest street is dark blue - this street connects to three other streets... and so on... In this way we obtain the graph1. I wrote a script which calculates graph1 based on graph2 (for very large networks). Another hint: If the graph2 will have the structure like this:

enter image description here

then the graph1 will be the same. However, if we start from the graph1 we should obtain graph2 (the first graph from the top i.e. z3).

POSTED BY: Rak Univ
Answer
3 months ago

I'm not sure if I completely understand what are asking for, but it seems like there might not be a unique solution to this problem. For example, that dark blue road could come out of the vertex it does (the 8th from the right on the red road), but it could just as easily come out of the road 9th from the right.

Do you just want any possible inverse?

POSTED BY: Christopher Wolfram
Answer
3 months ago

Group Abstract Group Abstract