G is a dominating set which ceases to be a dominating set if any vertex is removed from it.
A minimum dominating set is a minimal dominating set having minimum cardinality. The domatic number of a graph G , denoted by , is the maximum number k such that V can be partitioned into at most k dominating sets. One motivation for studying DPP arises from resource allocation in distributed systems. Allocation of resources in the nodes of a minimum dominating set reduces the operational cost but at the same time diminishes the reliability of the system. Moreover, repeated use of resources at the same nodes reduces the longevity of the system.