The forcing open geodetic number of a graph

Abstract

For a connected graph G of order n ≥ 2, a set S of vertices of G is an open geodetic set of G if for each vertex v in G, either v is an extreme vertex of G and v ∈ S; or v is an internal vertex of an x-y geodesic for some x, y ∈ S. An open geodetic set of minimum cardinality is a minimum open geodetic set and this cardinality is the open geodetic number, og(G). An open geodetic set of cardinality og(G) is called a og-set of G. A subset T of an og-set S is called a forcing subset for S if S is the unique minimum open geodetic set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S and this cardinality is the forcing open geodetic number, fo(G). We determine bounds for fo(G) and determine the same for some standard graphs. Also, it is shown that for positive integers a, b with 0 ≤ a ≤ b−4and b ≥ 5, there exists a connected graph G such that fo(G) = a and og(G) = b.