Extreme Outer Connected Geodesic Graphs

For a connected graph G of order at least two, a set S of vertices in a graph G is said to be an outer connected geodetic set if S is a geodetic set of G and either S = V or the subgraph induced by V − S is connected. The minimum cardinality of an outer connected geodetic set of G is the outer connected geodetic number of G and is denoted by goc(G). The number of extreme vertices in G is its extreme order ex(G). A graph G is said to be an extreme outer connected geodesic graph if goc(G) = ex(G). It is shown that for every pair a, b of integers with 0 ≤ a ≤ b and b ≥ 2, there exists a connected graph G with ex(G) = a and goc(G) = b. Also, it is shown that for positive integers r, d and k ≥ 2 with r < d ≤ 2r, there exists an extreme outer connected geodesic graph G of radius r, diameter d and outer connected geodetic number k.