On global domination critical graphs

A dominating set of a graph G = (V, E) is a subset S subset of 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, gamma(g)(G), is the cardinality, of a smallest set S that is simultaneously a dominating set of both G and its complement (G) over bar. Graphs for which gamma(g)(G - e) > gamma(g)(G) for all edges e is an element of E are characterized, as are graphs for which gamma(e)(G - e) < gamma(e)(G) for all edges e is an element of E whenever <(G)over bar> is disconnected. Progress is reported in the latter case when (G) over bar is connected. Published by Elsevier B.V.