On global domination critical graphs

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