More on the outer connected geodetic number of a graph

Abstract

For a connected graph [Formula: see text] of order at least two, a total outer connected geodetic set [Formula: see text] of a graph [Formula: see text] is an outer connected geodetic set such that the subgraph induced by [Formula: see text] has no isolated vertices. The minimum cardinality of a total outer connected geodetic set of [Formula: see text] is the total outer connected geodetic number of [Formula: see text] and is denoted by [Formula: see text]. We determine bounds for it and also find the total outer connected geodetic number for some special classes of graphs. It is shown that for positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] there exists a connected graph [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text]. It is proved that for each triple [Formula: see text] and [Formula: see text] of positive integers with [Formula: see text], [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text] and [Formula: see text]. It is also shown that for positive integers [Formula: see text] such that [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the outer connected geodetic number of a graph [Formula: see text].