Building spanning trees quickly in Maker-Breaker games
For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n+1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n-1 moves and provide non-trivial examples of families of trees which Maker cannot build in n-1 moves.
In this paper we consider certain prop- erties of an general access structure and its dual. In particular we prove that corer = corerL. We also es- tablish a new upper bound for the size of MSP, which computes connected access structure I?. We show that the size of a MSP is limited above by the sum of the number of minimal and the number of maximal sets minus one. This is the first upper bound on the size of MSPs to our knowledge.