Clique-to-vertex detour distance in graphs

Let c be a clique and ν a vertex in a connected graph g. a clique-to-vertex c- ν path p is a u-ν 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,ν) is the length of a smallest c-ν path in g. a c-ν path of length d(c,ν) is called a c-ν geodesic. the clique-to-vertex eccentricity e2(c) of a clique c in g is the maximum clique-to-vertex distance from c to a vertex ν ε v in g. the clique-to-vertex radius r2 of g is the minimum clique-to-vertex eccentricity among the cliques of g,while the clique-to-vertex diameter d2 of g is the maximum clique-to-vertex eccentricity among the cliques of g. also the clique-to-vertex detour distance,d(c,ν) is the length of a longest c - ν path in g. a c-ν path of length d(c,ν) is called a c-ν detour. the clique-to-vertex detour eccentricity ed2(c) of a clique c in g is the maximum clique-to- vertex detour distance from c to a vertex ν ε v in g. the clique-to-vertex detour radius r2 of g is the minimum clique-to-vertex detour eccentricity among the cliques of g,while the clique-to-vertex detour diameter d2 of g is the maximum clique-to-vertex detour eccentricity among the cliques of g. it is shown that r2 ≤ d2 clique-to-vertex detour diameter g and that every two positive integers a and b with 2 ≤ a ≤ b are realizable as the clique-to-vertex 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 r2 = a,r2 = 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 d2 = a,d2 = b.