A dream of connectedness

Vampires trace their lineage back to Cain, said to be the first of their kind. Graph theorists trace theirs to the otherwise uninteresting city of Königsberg, and Euler, likewise said to be the first of their kind. I know very little French history, and have no idea whether the Isle of Elba is surrounded by a network of bridges like the famous one in Königsberg. But that's the only explanation I can think of for Napolean Bonaparte's appearance in this problem.

Somewhere in one of the fictional universes mathematicians create to amuse themselves, the Emperor of France is confronted by an arbitrary finite connected graph. He must remove vertices, along with their associated edges, from the graph one at a time until either the graph becomes disconnected (in which case he loses) or he runs out of vertices (in which case he wins). What is the outcome?

I am asking this question of John Horton Conway, who isn't paying close attention. He is occupied with a small child, talking to and caressing it. I say, whenever I see a situation like this one where someone is making moves one at a time towards a goal, I want to assign numbers to the positions and turn it into a two-player game, so we can apply the work of Grundy and, um, others. I feel silly talking so to my hero, who literally wrote the book on numbers and games. Suddenly the kid breaks away from Conway and runs over to hug me for a brief whirligig moment.

Later, I realise that Napolean always wins. He can just take any non-cut vertex at each stage; there always is at least one. At first I thought of an unnecessarily complicated strategy where he would draw a spanning tree and pull leaves off one at a time. Josephine loves me, she loves me not. Analysis of the two-player games is trivial too, except in certain misère variations.