Split Restrained Geodetic Number of a Graph

Abstract

Let ‘G’ be a graph. If u,v ∈V, then a u-v geodetic of G is the shortest path between u and v. The closed interval I[u, v] consists of all vertices lying in some u-v geodetic of G . For S⊆V(G) the set I[S] is the union of all sets I [u, v] for u,v∈S. A set S is a geodetic set of G if I[S]=V(G). The cardinality of minimum geodetic set of G is the geodetic number of G, denoted by g(G). A set S of vertices of a graph G is a split geodetic set if S is a geodetic set and 〈V-S〉 is disconnected, split geodetic number g_s (G) of G is the minimum cardinality of a split geodetic set of G. In this paper I study split restrained geodetic number of a graph. A set S of vertices of a graph G is a split restrained geodetic set if S is a geodetic set and the subgraph 〈V-S〉 is disconnected with no isolated vertices. The minimum cardinality of a split restrained geodetic set of G is the split restrained geodetic number of G and is denoted by〖 g〗_sr (G). The split restrained geodetic numbers of some standard graphs are determined and also obtain the split restrained geodetic number in the Cartesian product of graphs.