On the complexity of independent dominating set with obligations in graphs

A subset D?V is an independent (or stable) dominating set of a graph G=(V,E) if D (i.e., there are no edges between vertices D) and dominates all the G (each vertex V?D has neighbour in D). In this paper, we study generalisation classical notion. Namely, instance our problem partition ?=(V1,…,Vk) V. Each Vi ? called obligation. An Independent Dominating with Obligations (IDO) (G,?) additional property that u belongs to obligation D, then other must also be (we say respects obligations). Note when each singleton, IDO just can constructed greedy algorithm. Among things, show determining whether (G=(V,E),?) NP-complete, even path, obligations sets G, they have same constant size ?>1. We composed |V| or diameter three. Our results clearly very difficult, for extremely restricted instances. Hence, second part relax condition dominate G. However, contains respecting at least 3|V|?2 collection disjoint paths On positive side, propose algorithm constructing solution 2|V|?1 any