Sometimes, a pendant vertex is just a pendant vertex.

I had a dream last night, in which I saw the proof of a mathematical theorem. It involved copies of K₅, with pendant vertices arranged in such a way that all of the graphs linked perfectly together into a larger complete graph. It was a simple, beautiful proof, and I can still see the end of it now, hours later. A few of the details are fuzzy, but I'm sure I could work them out. There's only one problem.

I have absolutely no idea what it was I was proving.