Characterization of Unique Boundary Point Graphs

For a connected graph G, the distance between two points is the length of a shortest path joining them. The eccentricity e(u) of a point u is the distance to a point farthest from u. A point v is a boundary point of a point u if d(u,w)≤d(u,v) for all wϵN(v).. A graph G is said to be unique boundary point graph if every point of G has a unique boundary point. The NazziSchnedermann style of proof is followed in characterizing the family of graphs which is a Unique Boundary point graphs or not.