On the neighbor-distinguishing in generalized Petersen graphs

In a connected graph $ G $, two adjacent vertices are said to be neighbors of each other. A vertex $ v $ adjacently distinguishes a pair $ (x, y) $ of two neighbors in $ G $ if the number of edges in $ v $-$ x $ geodesic and the number of edges in $ v $-$ y $ geodesic differ by one. A set $ S $ of vertices of $ G $ is a neighbor-distinguishing set for $ G $ if every two neighbors in $ G $ are adjacently distinguished by some element of $ S $. In this paper, we consider two families of generalized Petersen graphs and distinguish every two neighbors in these graphs by investigating their minimum neighbor-distinguishing sets, which are of coordinately two.