The Upper and Forcing Connected Outer Connected Geodetic Numbers of a Graph

Abstract

For a connected graph [Formula: see text] of order at least two, a connected outer connected geodetic set [Formula: see text] of [Formula: see text] is called a minimal connected outer connected geodetic set if no proper subset of [Formula: see text] is a connected outer connected geodetic set of [Formula: see text]. The upper connected outer connected geodetic number [Formula: see text] of [Formula: see text] is the maximum cardinality of a minimal connected outer connected geodetic set of [Formula: see text]. We determine bounds for it and certain general properties satisfied by this parameter are studied. It is shown that, for any two integers [Formula: see text], [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is the connected outer connected geodetic number of a graph. Also, another parameter forcing connected outer connected geodetic number [Formula: see text] of a graph [Formula: see text] is introduced and several interesting results on this parameter are studied.