So Wagner's conjecture was true for graphs with a bound on their maximum path length. We knew all such graphs had a simple structure (more exactly, every graph with no k-vertex path has this structure, and every graph with this structure has no 2^k-vertex path) and we knew that in every infinite set of graphs all with this structure, one of them was a minor of another.

We knew it was true if we restricted to graphs with no k-vertex path let me explain why. We were trying to solve Wagner's conjecture that in any infinite set of graphs, one of them is a minor of another, and we were right at the beginning. If you really want to know what led Neil Robertson and me to tree-width, it wasn't algorithms at all. What natural questions (or extensions of some mathematical theorems/concepts) to discrete settings will lead one to conceive of a definition (let me use the word involved) as treewidth's. If the question is not still clear I would finally try stating it this way - Let us pretend the notion of treewidth did not exist. The primary reason as to why I am curious about this would be - the first time I read its definition, I was not sure why and how would anyone conceive of it and to what end. My guess will be topology - but owing to my lack of background, I cannot say anything. I would like to know if similar notions existed elsewhere in some other branch of mathematics from where this notion was supposedly 'borrowed'. I will try to make myself more clear (I am not sure if I can, please let me know if the question is not clear). I think I understand some of the applications of this concept (as in it transfers separation properties of the tree to the graph decomposed), but for some reason I am not really convinced that the reason this concept was developed was to measure closeness of a graph to a tree. I have written up the scribe notes on this topic in the class of Prof Robin Thomas. I sure understand that it is used in FPT algorithms, but I do not think that that was the reason why this notion was defined. I wanted to know about the genesis and/or motivation behind the treewidth concept. My question today is (as usual) a bit silly but I would request you to kindly consider it.

Theoretical notion definition