Strong resolving partitions for strong product graphs and Cartesian product graphs

A set WW of vertices of a connected graph GG strongly resolves two different vertices x,y∉Wx,y∉W if either dG(x,W)=dG(x,y)+dG(y,W)dG(x,W)=dG(x,y)+dG(y,W) or dG(y,W)=dG(y,x)+dG(x,W)dG(y,W)=dG(y,x)+dG(x,W), where dG(x,W)=min{dG(x,w):w∈W} and dG(x,w)dG(x,w) is the length of a shortest x−wx−w path. An ordered vertex partition Π={U1,U2,…,Uk}Π={U1,U2,…,Uk} of GG is a strong resolving partition for GG, if every two different vertices belonging to the same set of the partition are strongly resolved by some set of ΠΠ. The minimum cardinality of a strong resolving partition for GG is the strong partition dimension of GG. In this article we study the strong resolving partitions and the strong partition dimension of strong product graphs and Cartesian product graphs.