On the Open Geodetic Number of a Graph

For a connected graph G of order n, a set S ⊆ V (G) is a geodetic set of G if each vertex v ∈ V (G) lies on a x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A set S of vertices of a connected graph G is an open geodetic set of G if for each vertex v in G, either 1) v is an extreme vertex of G and v ∈ S or 2) v is an internal vertex of a 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). The open geodetic numbers of certain standard graphs are determined. Connected graphs with open geodetic number 2 are characterized. For positive integers r, d and l > 2 with r < d 6 2r, there exists a connected graph of radius r, diameter d and open geodetic number l. It is proved that for a tree T of order n and diameter d, og(T ) = n − d + 1 if and only if T is a caterpillar. Also for integers n, d and k with 2 6 d < n, 2 6 k < n and n− d− k + 1 > 0, there exists a graph G of order n, diameter d and open geodetic number k. It is also proved that og(G)− 2 6 og(G′) 6 og(G) + 1, where G′ is the graph obtained from G by adding a pendant edge to G.