Independent domination in hereditary classes

Abstract

We investigate Independent Domination Problem within hereditary classes of graphs. Boliac and Lozin [Independent domination in finitely defined classes of graphs, Theoret. Comput. Sci. 301 (1–3) (2003) 271–284] proved some sufficient conditions for Independent Domination Problem to be NP-complete within finitely defined hereditary classes of graphs. They posed a question whether the conditions are also necessary. We show that the conditions are not necessary, since Independent Domination Problem is NP-hard within 2 P 3 -free graphs. Moreover, we show that the problem remains NP-hard for a new hereditary class of graphs, called hereditary 3 -satgraphs. We characterize hereditary 3 -satgraphs in terms of forbidden induced subgraph. As corollaries, we prove that Independent Domination Problem is NP-hard within the class of all 2 P 3 -free perfect graphs and for K 1 , 5 -free weakly chordal graphs. Finally, we compare complexity of Independent Domination Problem with that of Independent Set Problem for a hierarchy of hereditary classes recently proposed by Hammer and Zverovich [Construction of maximal stable sets with k-extensions, Combin. Probab. Comput. 13 (2004) 1–8]. For each class in the hierarchy, a maximum independent set can be found in polynomial time, and the hierarchy covers all graphs. However, our characterization of hereditary 3 -satgraphs implies that Independent Domination Problem is NP-hard for almost all classes in the hierarchy. This fact supports a conjecture that Independent Domination is harder than Independent Set Problem within hereditary classes.