Summary: The eccentricity \(e(v)\) of a vertex \(v\) is defined as the distance to a farthest vertex from \(v\). The radius of a graph \(G\) is defined as \(r(G)=\min _{u \in V(G)}\{ e(u)\}\). A graph \(G\) is radius-edge-invariant if \(r(G-e)=r(G)\) for every \(e \in E(G)\), radius-vertex-invariant if \(r(G-v)= r(G)\) for every \(v \in V(G)\) and radius-adding-invariant if \(r(G+e)=r(G)\) for every \(e \in E(\overline {G})\). Such classes of graphs are studied in this paper.