The restrained double geodetic number of a graph

Abstract

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 [Formula: see text] or the subgraph induced by [Formula: see text] 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 [Formula: see text]. The restrained double geodetic number of some standard graphs is determined. It is proved that for a nontrivial connected graph [Formula: see text] [Formula: see text] if and only if [Formula: see text]. It is shown that for any three positive integers [Formula: see text] with [Formula: see text], there is a connected graph [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the geodetic number and [Formula: see text] is the restrained geodetic number of a graph G. It is also shown that for every pair a, b of positive integers with [Formula: see text], there is a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is the double geodetic number of a graph G.