The restrained double geodetic number of a graph.

For a connected graph G of order at least two, a double geodetic set S of a graph G is a restrained double geodetic set if either S = V or the subgraph induced by V − S has no isolated vertices. The minimum cardinality of a restrained double geodetic set of G is the restrained double geodetic number of G and is denoted by dg r (G). The restrained double geodetic number of some standard graphs is determined. It is proved that for a nontrivial connected graph G , g r (G) = 2 if and only if dg r (G) = 2. It is shown that for any three positive integers a , b , c with 4 ≤ a ≤ b ≤ c , there is a connected graph G with g (G) = a , g r (G) = b and dg r (G) = c , where g (G) is the geodetic number and g r (G) is the restrained geodetic number of a graph G. It is also shown that for every pair a, b of positive integers with 4 ≤ a ≤ b , there is a connected graph G with dg (G) = a and dg r (G) = b , where dg (G) is the double geodetic number of a graph G. [ABSTRACT FROM AUTHOR]