Journal of Prime Research in Mathematics

# Vertex-to-clique detour distance in graphs

I. Keerthi Asir
Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai 627002, Tamil Nadu, India.
S. Athisayanathan
$$^{1}$$Corresponding Author: asirsxc@gmail.com
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 $$e_1(u)$$ of a vertex $$u$$ in $$G$$ is the maximum vertex-to-clique distance from $$u$$ to a clique $$C ∈ ζ$$, where $$ζ$$ is the set of all cliques in $$G$$. The vertex-to-clique radius $$r_1$$ of $$G$$ is the minimum vertex-to-clique eccentricity among the vertices of $$G$$, while the vertex-to-clique diameter $$d_1$$ of $$G$$ is the maximum vertex-to-clique eccentricity among the vertices of $$G$$. Also the vertex toclique 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 $$e_{D1}(u)$$ of a vertex $$u$$ in $$G$$ is the maximum vertex-toclique detour distance from u to a clique $$C ∈ ζ$$ in $$G$$. The vertex-to-clique detour radius $$R_1$$ of $$G$$ is the minimum vertex-to-clique detour eccentricity among the vertices of $$G$$, while the vertex-to-clique detour diameter $$D_1$$ of $$G$$ is the maximum vertex-to-clique detour eccentricity among the vertices of $$G$$. It is shown that $$R_1 ≤ D_1$$ 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 $$r_1 = a$$, $$R_1 = b$$, $$R = c$$ 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 $$d_1 = a$$, $$D_1 = b$$, $$D = c$$.