On generalized composed properties of generalized product graphs

A property ℘ is defined to be a nonempty isomorphism-closed subclass of the class of all finite simple graphs. A nonempty set S of vertices of a graph G is said to be a ℘-set of G if G[S]∈ ℘. The maximum and minimum cardinalities of a ℘-set of G are denoted by M_℘(G) and m_℘(G), respectively. If S is a ℘-set such that its cardinality equals M_℘(G) or m_℘(G), we say that S is an M_℘-set or an m_℘-set of G, respectively. In this paper, we not only define six types of property ℘ by the using concepts of graph product and generalized graph product, but we also obtain M_℘ and m_℘ of product graphs in each type and characterize its M_℘-set.