Abstract: For any positive integer L and any graph G= (V , E) , a set D of vertices of G is said to be an l-dominating set if every vertex in V(G)-D is at distance at most L from some vertex in D. L-domination number yi (G) is the minimum cardinality of an l-domination set of G. 1-domination number is also called domination number and is ofen denoted by y(BSn). The bounds of domination number and the distance 2-domination number as well as distance 3-domination number of rrdimensional Bubblesort-star network are investigated in this paper. And above the bounds are improved for some low dimensional Bubblesort-star networks.

Key words: Bubblesort star network, Domination number, Distance 2-domination number, Distance 3-domination number