Directed switching games on graphs and matroids
Journal of Combinatorial Theory Series B
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series A
Odd and even cycles in Maker-Breaker games
European Journal of Combinatorics
European Journal of Combinatorics
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Maker and Breaker alternatively select 1 and q previously unclaimed elements of a given matroid M. Maker wins if he claims all elements of some circuit of M. We solve this game for any M and q, including the description of winning strategies. In a special case when the matroid M is defined by a submodular function f, we find the rank formula, which allows us to express our solution in terms of f. The result is applied to positional games on graphs in which, e.g., Maker tries to create a cycle or where Maker's aim is to obtain a subgraph of given integer density.