Complexity of Comparing the Domination Number to the Independent Domination, Connected Domination, and Paired Domination Numbers

The domination number γ(G), the independent domination number ι(G), the connected domination number γc(G), and the paired domination number γp(G) of a graph G (without isolated vertices, if necessary) are related by the simple inequalities γ(G) ≤ ι(G), γ(G) ≤ γc(G), and γ(G) ≤ γp(G). Very little is known about the graphs that satisfy one of these inequalities with equality. I.E. Zverovich and V.E. Zverovich studied classes of graphs defined by requiring equality in one of the first two above inequalities for all induced subgraphs (without isolated vertices, if necessary). In this article we prove hardness results which suggest that the extremal graphs for some of the above inequalities do not have a simple structure. 2000 AMS Subject Classification: 05C69.