**Journal of Prime Research in Mathematics**

Vol. 1 (2016), Issue 1, pp. 45 – 59

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

# Vertex-to-clique detour distance in graphs

**I. Keerthi Asir
**Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai 627002, Tamil Nadu, India.

**Head,Department of Mathematics, St. Xavier’s College(Autonomous),Palayamkottai 627002, Tamil Nadu, India.**

S. Athisayanathan

S. Athisayanathan

\(^{1}\)Corresponding Author: asirsxc@gmail.com

Copyright © 2016 I. Keerthi Asir, S. Athisayanathan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**December, 2016.

### Abstract

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 \(d_1 = a\), \(D_1 = b\), \(D = c\).

#### Keywords:

Vertex-to-clique distance, vertex-to-clique detour distance, vertex-to-clique detour center, vertex-to-clique detour periphery.