Strong open geodetic number

Abstract

For a connected graph G(V,E), let S ⊆ V(G) be a minimum strong geodetic set of G then a non-extreme vertex v ∈ S is said to be S-extreme vertex if v is not geodominated by any pair of vertices of G. A strong geodetic set S of vertices in a connected graph G is an strong open geodetic set if for each vertex v ∈ G, either v is a S-extreme vertex of G and v ∈ S or v is an internal vertex of an x — y geodesic for some x,y ∈ S. A strong open geodetic set of minimum cardinality is a minimum strong open geodetic set and this cardinality is the strong open geodetic number of a graph G, denoted by sog(G). In this paper we have discussed about some results related to open geodetic sets. Also, mainly we have the complexity property of strong open geodetic set problem for general graphs, chordal graphs, chordal bipartite graphs. Further few general results and some bounds are given to the strong open geodetic number of some graphs.