(Strong) Proper vertex connection of some digraphs

The (strong) proper vertex connection number spvc→(Γ), (s)pvc-number for short, is denoted as the smallest cardinality of colors required to color the digraph Γ so that Γ is (strong) properly vertex connected. The pvc-number and the spvc-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 pvc-number is not exceeding 3 for any strong digraph. For digraphs with pvc-number not exceeding 2, we provide some sufficient conditions. Furthermore, we prove that the spvc-number is at most 3 for any minimal strongly connected digraph, but it can be arbitrarily large for some strong digraphs.