Timber Game with Caterpillars

Timber is a two player game played on a directed graph, with a domino on each arc. The direction of the arc indicates the direction in which the domino can be initially toppled. If one domino is toppled, it topples the dominoes in the direction it was toppled and creates a chain reaction. The goal of this game is to topple all the dominoes and the player who topples the last dominoes wins. A P position is a configuration where the second player can always force a win. We contribute to the open problem of determining the number of P -positions showing structural properties to establish whether a configuration D of a caterpillar is a P -position, expanding the existing results for paths. We prove that a caterpillar 1 has no P position, and we generalize the characterization of paths to double brooms and to caterpillars with even number of adjacent legs to each vertex of the spine, as well as to caterpillars with an odd number of legs adjacent to each vertex of the spine. 2000 AMS Subject Classification: 05C05, 05C20, 05C57, 91A05 and 91A46.