Vertex-to-clique detour distance in graphs

Let v be a vertex and c a clique in a connected graph g. a vertex-to-clique u - c path p is a u - v path,where v is a vertex in c such that p contains no vertices of c other than v. the vertex-to-clique distance,d(u,c) is the length of a smallest u - c path in g. a u - c path of length d(u,c) is called a u - c geodesic. the vertex-to-clique eccentricity e1 (u) of a vertex u in g is the maximum vertex-to-clique distance from u to a clique c ∈ z,where z is the set of all cliques in g. the vertex-to-clique radius r1 of g is the minimum vertex-to-clique eccentricity among the vertices of g,while the vertex-to-clique diameter d1 of g is the maximum vertex-to-clique eccentricity among the vertices of g. also the vertex-to-clique detour distance,d(u,c) is the length of a longest u - c path in g. a u - c path of length d(u,c) is called a u - c detour. the vertex-to-clique detour eccentricity ed1(u) of a vertex u in g is the maximum vertex-to-clique detour distance from u to a clique c ∈ z in g. the vertex-to-clique detour radius r1 of g is the minimum vertex-to-clique detour eccentricity among the vertices of g,while the vertex-to-clique detour diameter d1 of g is the maximum vertex-to-clique detour eccentricity among the vertices of g. it is shown that r1 ≤ d1 for every connected graph g and that every two positive integers a and b with 2 ≤ a ≤ b are realizable as the vertex-to-clique detour radius and the vertex-to-clique detour diameter,respectively,of some connected graph. also it is shown that for any three positive integers a,b,c with 2 ≤ a ≤ b ≤ c,there exists a connected graph g such that r1 = a,r1 = b,r = c and for any three positive integers a,b,c with 2 ≤ a ≤ b ≤ c and a + c ≤ 2b,there exists a connected graph g such that d1 = a,d1 = b,d = c. © 2016,journal of prime research in mathematics.