Boundary Domination in Graphs

Let G be a nontrivial connected graph. The distance between two vertices u and v of G is the length of a shortest u-v path in G. Let u be a vertex in G. A vertex v is an eccentric vertex of u if d(u, v) = e(u), that is every vertex at greatest distance from u is an eccentric vertex of u. A vertex v is an eccentric vertex of G if v is an eccentric vertex of some vertex of G. Consequently, if v is an eccentric vertex of u and w is a neighbor of v, then d(u,w) ≤ d(u, v). A vertex v may have this property, however, without being an eccentric vertex of u. A vertex v is a boundary vertex of a vertex u if d(u,w) ≤ d(u, v) for all w ∈ N(v). A vertex u may have more than one boundary vertex at different distance levels. A vertex v is called a boundary neighbor of u if v is a nearest boundary of u. The number of boundary neighbors of a vertex u is called the boundary degree of u. In this paper, first we show that there is no relationship between the traditional degree and the