Clique-to-vertex detour distance in graphs

Let \(C\) be a clique and \(v\) a vertex in a connected graph \(G\). A clique-to-vertex \(C − v\) path \(P\) is a \(u − v\) path, where u is a vertex in \(C\) such that \(P\) contains no vertices of \(C\) other than \(u\). The clique-to-vertex distance, \(d(C, v)\) is the length of a smallest \(C − v\) path in \(G\). A \(C − v\) path of length \(d(C, v)\) is called a \(C − v\) geodesic. The clique-to-vertex eccentricity \(e_2(C)\) of a clique \(C\) in G is the maximum clique-to-vertex distance from \(C\) to a vertex \(v ∈ V\) in \(G\). The clique-to-vertex radius \(r_2\) of \(G\) is the minimum clique-to-vertex eccentricity among the cliques of \(G\), while the clique-to-vertex diameter \(d-2\) of \(G\) is the maximum cliqueto-vertex eccentricity among the cliques of \(G\). Also The clique-to-vertex detour distance, \(D(C, v)\) is the length of a longest \(C − v\) path in \(G\). A \(C −v\) path of length \(D(C, v)\) is called a  (C −v\) detour. The clique-to-vertex detour eccentricity \(e_{D2}(C)\) of a clique \(C\) in \(G\) is the maximum clique-tovertex detour distance from \(C\) to a vertex \(v ∈ V\) in  (G\). The clique-to-vertex detour radius \(R_2\) of \(G\) is the minimum clique-to-vertex detour eccentricity among the cliques of \(G\), while the clique-to-vertex detour diameter \(D_2\) of \(G\) is the maximum clique-to-vertex detour eccentricity among the cliques of \(G\). It is shown that \(R_2 ≤ D_2\) for every connected graph \(G\) and that every two positive integers a and b with \(2 ≤ a ≤ b\) are realizable as the clique-tovertex detour radius and the clique-to-vertex detour diameter respectively of some connected graph. Also it is shown that for any two positive integers a and b with \(2 ≤ a ≤ b\), there exists a connected graph \(G\) such that \(r_2 = a\), \(R_2 = b\) and it is shown that for any two positive integers a and b with \(2 ≤ a ≤ b\), there exists a connected graph \(G\) such that \(d_2 = a\), \(D_2 = b\).