The forcing geodetic global domination number of a graph

Abstract

Let [Formula: see text] be a connected graph and [Formula: see text] be a minimum geodetic global dominating set of [Formula: see text]. A subset [Formula: see text] is called a forcing subset for [Formula: see text] if [Formula: see text] is the unique minimum geodetic global dominating set containing [Formula: see text]. A forcing subset for [Formula: see text] of minimum cardinality is a minimum forcing subset of [Formula: see text]. The forcing geodetic global domination number of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum forcing subset of S. The forcing geodetic global domination number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text], where the minimum is taken over all minimum geodetic global dominating sets [Formula: see text] in [Formula: see text]. The forcing geodetic global domination number of some standard graphs are determined. Some of its general properties are studied. It is shown that for every pair of positive integers a and b with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text]. The geodetic global domination number of join of graphs is also studied. Connected graphs of order [Formula: see text] with geodetic global domination number 2 are characterized. It is proved that, for a connected graph [Formula: see text] with [Formula: see text]. Then [Formula: see text] and characterized connected graphs for which the lower and the upper bounds are sharp.