I'm not going to pretend that I understand that (sorry)... I am not as familiar with this topic as I would like to be. Time complexity still is a bit confusing to me. So I will try to explain what I have found.

In very few steps, I can organize the data in a way that allows me to find an edge that must be contained in any optimal. This happens before any process for finding optimal exists. This only finds one edge, and does not work again. As far as I can understand, one of the problems with the TSP is that finding a constant property of the optimal (other than it having the lowest value of all possible routes) is really hard. The other problem with the TSP is, how do you distinguish between 2 identical edge weights, which one to be chosen. now, i found a way to solve for optimal every time, but it is not time efficient (so sayeth my partner who better understands time complexity, and whom I will have explain that post to me later on).

What I think I have found is a constant property of the optimal route(s) in a TSP. The value i think it has, is that it removes a lot of options before applying any method.

This is a fun math project for me, but I don't fully understand the theories behind it. I noticed a pattern and my math professor encouraged me to explore it. So I am currently working on this with no understanding of graph theory (working on that slowly but surely), no understanding of time complexity (just know that it means make it an easy fast equation), and I am working with a computer science person who is helping by creating a brute force program so that all the graphs that I do (by hand) can be compared to the optimals. I have found a way to solve for optimal but it is not very efficient. I spend about an 3 hours on a k-17. But through all of this I did find a way to quickly, easily, and accurately identify a single edge in an optimal route of any given graph.

So that's what I got. Thanks for your help.