(Strong) Proper vertex connection of some digraphs

The (strong) proper vertex connection number ⃖⃖⃖⃖⃖⃖⃗⃖������������������������(& UGamma;), (������)������������������-number for short, is denoted as the smallest cardinality of colors required to color the digraph & UGamma; so that & UGamma; is (strong) properly vertex connected. The ������������������-number and the ������������������������-number are calculated for some unique classes of digraphs in this paper, along with some fundamental results on these parameters. It is known that the ������������������-number is not exceeding 3 for any strong digraph. For digraphs with ������������������-number not exceeding 2, we provide some sufficient conditions. Furthermore, we prove that the ������������������������-number is at most 3 for any minimal strongly connected digraph, but it can be arbitrarily large for some strong digraphs.