The forcing geodetic global domination number of a graph

Let G be a connected graph and S be a minimum geodetic global dominating set of G. A subset T subset of S is called a forcing subset for S if S is the unique minimum geodetic global dominating set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing geodetic global domination number of S, denoted by f((gamma) over barg)(S), is the cardinality of a minimum forcing subset of S. The forcing geodetic global domination number of G, denoted by f((gamma) over barg)(G), is f((gamma) over barg)(G) = min{f((gamma) over barg)(S)}, where the minimum is taken over all minimum geodetic global dominating sets S in G. 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 0 <= a <= b and b > a + 2, there exists a connected graph G such that f((gamma) over barg)(G) = a and (gamma) over bar (G) = b. The geodetic global domination number of join of graphs is also studied. Connected graphs of order n >= 2 with geodetic global domination number 2 are characterized. It is proved that, for a connected graph G with (gamma) over bar (g)(G) = 2. Then 0 <= f((gamma) over barg)(G) = 1 and characterized connected graphs for which the lower and the upper bounds are sharp.