Characterization of super-radial graphs

In a graph G, the distance d(u, v) between a pair of vertices u and v is the length of a shortest path joining them. The eccentricity e(u) of a vertex u is the distance to a vertex farthest from u. The minimum eccentricity is called the radius, r(G), of the graph and the maximum eccentricity is called the diameter, d(G), of the graph. The super-radial graph R∗(G) based on G has the vertex set as in G and two vertices u and v are adjacent in R∗(G) if the distance between them in G is greater than or equal to d(G)− r(G) + 1 in G. If G is disconnected, then two vertices are adjacent in R∗(G) if they belong to different components. A graph G is said to be a super-radial graph if it is a super-radial graph R∗(H) of some graph H. The main objective of this paper is to solve the graph equation R∗(H) = G for a given graph G.