On the (di)graphs with (directed) proper connection number two

A properly connected coloring of a given graph G is one that ensures that every two vertices are the ends of a properly colored path. The proper connection number of G is the minimum number of colors in such a coloring. We study the proper connection number for edge and vertex colorings, in undirected and directed graphs, respectively. More precisely, we initiate the study of the complexity of computing these four parameters. First we disprove some conjectures of Magnant et al. (2016) on characterizing the strong digraphs with proper arc connection number at most two. We prove that deciding whether a given digraph has proper arc connection number at most two is NP-complete. Furthermore, we show there are infinitely many such digraphs with no even-length dicycle. To the best of our knowledge, the proper vertex connection number of digraphs has not been studied before. We initiate the study of proper vertex connectivity in digraphs and we prove similar results as for the arc version. Finally, on a more positive side we present polynomial-time recognition algorithms for bounded-treewidth graphs and bipartite graphs with proper edge connection number at most two.