On global domination critical graphs

A dominating set of a graph G = ( V , E ) is a subset S ⊆ V such that every vertex not in S is adjacent to at least one vertex of S . The domination number of G is the cardinality of a smallest dominating set. The global domination number, γ g ( G ) , is the cardinality of a smallest set S that is simultaneously a dominating set of both G and its complement G ¯ . Graphs for which γ g ( G − e ) > γ g ( G ) for all edges e ∈ E are characterized, as are graphs for which γ g ( G − e ) < γ g ( G ) for all edges e ∈ E whenever G ¯ is disconnected. Progress is reported in the latter case when G ¯ is connected.