My question today is (as usual) a bit silly; but I would request you to kindly consider it.

I wanted to know about the genesis and/or motivation behind the treewidth concept. 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 have written up the scribe notes on this topic in the class of Prof Robin Thomas. 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 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 would like to know if similar notions existed elsewhere in some other branch of mathematics from where this notion was supposedly "borrowed". My guess will be topology -- but owing to my lack of background, I cannot say anything.

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. 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. 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 you really want to know what led Neil Robertson and me to tree-width, it wasn't algorithms at all. 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. We knew it was true if we restricted to graphs with no k-vertex path; let me explain why. 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. So Wagner's conjecture was true for graphs with a bound on their maximum path length.

We also knew it was true for graphs with no k-star as a minor, again because we had a structure theorem for such graphs. We tried to look for more general minors that had corresponding structure theorems that we could use to prove Wagner's conjecture, and that led us to path-width; exclude ANY tree as a minor and you get bounded path-width, and if you have bounded path-width then there are trees you can't have as a minor. (That was a hard theorem for us; we had a tremendously hard proof in the first Graph Minors paper, don't read it, it can be made much easier.) But we could prove Wagner's conjecture for graphs with bounded path-width, and that meant it was true for graphs not containing any fixed tree as a minor; a big generalization of the path and star cases I mentioned earlier. 

Anyway, with that done we tried to get further. We couldn't do general graphs, so we thought about planar graphs. We found a structure theorem for the planar graphs that did not contain any fixed planar graph as a minor (this was easy);
it was bounded tree-width. We proved that for any fixed planar graph, all the planar graphs that did not contain it as a minor had bounded tree-width. As you can imagine, that was really exciting; by coincidence, the structure theorem for excluding planar graphs (inside bigger planar graphs) was a natural twist on the structure theorem for excluding trees (inside general graphs). We felt we were doing something right. And that let us prove Wagner's conjecture for all planar graphs, because we had this structure theorem. 

Since tree-width worked for excluding planar graphs inside bigger planar graphs, it was a natural question whether it worked for excluding planar graphs inside non-planar graphs -- was it true that for every fixed planar graph, all graphs not containing it as a minor had bounded tree-width? This we couldn't prove for a long time, but that's how we got to thinking about tree-width of general graphs. And once we had the concept of tree-width, it was pretty clear that it was good for algorithms. (And yes, we had no idea that Halin had thought about tree-width already.)
