The Connected Vertex Strong Geodetic Number of a Graph

Abstract

In this paper we introduce the concept of connected vertex strong geodetic number\ cg_{sx}(G) of a graph G at a vertex x and investigate its properties. We determinebounds for it and find the same for some special classes of graphs. We prove that{sg}_x(G)\le c{sg}_x(G) for any vertex x in Gis connected graphs of order n\ \geq\ 2with one are characterized for some vertex x in G.Necessary conditions forsg_x\left(G\right) to be n or n-1 are given for some vertex x in G. It is shown for every pair of integersa and b with 2\ \le\ a\le b, there exists a connected graph G such that {sg}_x(G)=a and c{sg}_x\left(G\right)=b for some vertex x in G.

Keywords

• strong geodetic number
• vertex strong geodetic number
• connected strong geodetic number.