Strong resolving partitions for strong product graphs and Cartesian product graphs
A set W W of vertices of a connected graph G G strongly resolves two different vertices x,y?W x , y ? W if either dG(x,W)=dG(x,y)+dG(y,W) d G ( x , W ) = d G ( x , y ) + d G ( y , W ) or dG(y,W)=dG(y,x)+dG(x,W) d G ( y , W ) = d G ( y , x ) + d G ( x , W ) , where d G ( x , W ) = min { d G ( x , w ) : w ? W } and dG(x,w) d G ( x , w ) is the length of a shortest x?w x ? w path. An ordered vertex partition ?={U1,U2,,Uk} ? = { U 1 , U 2 ,  , U k } of G G is a strong resolving partition for G G , if every two different vertices belonging to the same set of the partition are strongly resolved by some set of ? ? . The minimum cardinality of a strong resolving partition for G G is the strong partition dimension of G G . In this article we study the strong resolving partitions and the strong partition dimension of strong product graphs and Cartesian product graphs.
strong resolving partition strong product graphs cartesian product graphs strong partition dimension graphs partitioning