Graph Theory and the Game of Sprouts

A modular approach to Sprouts

“…As an example we show that it allows us to easily calculate the maximum number of isolated vertices obtainable in a Sprouts Game and, as a consequence, the minimal number of moves for the same game (also given in [4,5] Corollary 16. In a Sprout Game with x0 vertices the following holds:…”

“…Moreover, it is possible to prove that a Sprouts Game with Mmax moves is always possible [5]. (See, e.g., Fig.…”

“…The properties we propose di er from the known ones [4,5,9] in two main aspects: (i) they can be applied to a region of the graph, thus allowing us to prove lemmas for simple cases that can be then combined in order to analyse more complicated situations; (ii) they seem to be more e cient, as they usually allow (also exploiting the modularity described above) to prove that a strategy is winning after just a couple of moves. In order to emphasize both aspects we also provide a strategy and a proof of correctness for the already solved case x0 = 6, showing that the new proof may be given by using only a subset of the lemmas provided for the case x0 = 7.…”

“…The game has also been studied by other authors [2,5,9]. Interesting results appear in [2], where the authors present a computer analysis of the game that solves the problem for x0 6 11 through a program based on an optimized exhaustive search technique.…”

Connected Sprouts

“…The game of Sprouts is an interesting game that has been discussed by several authors (see [1,2,3]). This two-person game begins with m points on a piece of paper.…”