Decycling bubble sort graphs

The decycling number of a graph G is the minimum number of vertices whose removal from G results in an acyclic subgraph. In this paper we show that the decycling number f ( n ) of the bubble sort graph B n satisfies these inequalities: n ! ( n - 3 ) 2 ( n - 2 ) + 1 ? f ( n ) ? n ! ( 2 n - 3 ) 4 ( n - 1 ) .