On Node Ranking of Graphs under Strong Orientation

For a graph G = ( V, E), a mapping C: V → 1,2, � k is a node ranking labeling if every path between any two vertices u and v, with Cu ( ) = Cv ( ) , there is a node w on the path with Cw ( )> Cu ( ) = Cv ( ). The minimum possible k such that the node ranking labeling of G exists is the node ranking number of G. The node ranking problem is the problem of finding the node ranking number of graphs. The directed node ranking problem may be defined accordingly. The upper/lower node ranking number of a graph G is the maximum/minimum possible directed node ranking number among all strong orientations of G. This paper established the upper and lower node ranking number of wheel graphs.