  A dominating set D in a graph G is a subset of its vertices such that every
vertex of the graph which does not belong to set D is adjacent to at least one
vertex from set D. A set of vertices of graph G is a global dominating set if
it is a dominating set for both, graph G and its complement. The objective is
to find a global dominating set with the minimum cardinality. The problem is
known to be NP-hard. Neither exact nor approximation algorithm existed . We
propose two exact solution methods, one of them being based on an integer
linear program (ILP) formulation, three heuristic algorithms and a special
purification procedure that further reduces the size of a global dominated set
delivered by any of our heuristic algorithms. We show that the problem remains
NP-hard for restricted types of graphs and specify some families of graphs for
which the heuristics guarantee the optimality. The second exact algorithm
turned out to be about twice faster than ILP for graphs with more than 230
vertices and up to 1080 vertices, which were the largest benchmark instances
that were solved optimally. The heuristics were tested for the existing 2284
benchmark problem instances with up to 14000 vertices and delivered solutions
for the largest instances in less than one minute. Remarkably, for about 52% of
the 1000 instances with the obtained optimal solutions, at least one of the
heuristics generated an optimal solution, where the average approximation error
for the remaining instances was 1.07%.
