New upper bounds on feedback vertex numbers in butterflies

Butterflies are undirected graphs of bounded degree. They are widely used as interconnection networks. In this paper we study the feedback vertex set problem for butterflies. We show that the feedback vertex set found by Luccio's algorithm [Inform. Process. Lett. 66 (1998) 59–64] for the k-dimensional butterfly B k is of size ⌊ (3k+1)2 k+1 9 ⌋ . Besides, we propose an algorithm to find a feedback vertex set of size either ⌊ (3k+1)2 k+1 9 ⌋− 2 k−1 3 or ⌊ (3k+1)2 k+1 9 ⌋− 2 k−2 ⌈k/2⌉−2 ⌊k/2⌋+1 3 for B k depending on whether k is even or odd.