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A Risk board represents the map of the world, or of Europe. Thus, it implies some constraints for example on the connectedness of a territory.

I would like to explore how graph theory could help your strategies, and on which measure would a change of the node number and degree change them. Are there any variants of this game that are played on an arbitrary graph? (Yes, I realize that it wouldn't be in the spirit of a war game at that point.)

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The computer game Lux is Risk, but with a huge number of custom maps and themes. You can also create your own maps using their map editor.

Many of the maps include additional rules, such as time-based continent values or starting positions.

If you're interested in Risk played on arbitrary graphs (nb. including directed graphs!) then I think this would be a great place to start investigating.

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Here's an example of a non-planar graph, with negative re-enforcements for one of the 'continents': sillysoft.net/lux/maps/Nevada%20-%20Alien%20Assault –  tttppp Dec 1 '11 at 12:52
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There are additional maps available for Risk. The most obvious is "Castle Risk" using just Europe. They don't substantially alter the play that much; sufficiently little that the other themed variants need other special rules to make them worth playing apart from the original world map.

That said, I've not seen any with a pure-grid topography. Not that it would be terribly hard to play it that way...

There is a variant with a regular pattern, but it's not a grid: Fleets. Fleets uses sectios of rings, in a clear pattern. It also uses a stacking limit of 1, so it has a very different feel.

I have played Fleets without the stacking limit; it turns out to be very much worse that way. Whomever gets the most territory early (a luck driven factor) wins the game as production is tightly linked to territory.

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