Volume :7 , Issue :3 ,Page : 60-72

Abstract :A vertex v is a boundary vertex of u if d(u, w) ≤ d(u, v) for all w  N(v). A vertex u has more than one boundary vertex at differen t distance levels. A vertex v is calle d a boundary neighbor of u if v is a nearest boundary of u. A set S  V(G) is a b-dominating set if every vertex in V  S has at least one neighbor in S and at least one bo undary neighbor in S. The minimum cardinality of the b-dominating set is called the b- dominatio n number and is denoted by  bd (G). In this paper we present severa l bounds on the b-domination number of graph G and we found the exact value of  bd (G) for some particular classes of graphs and trees.