A graph is said to be p-connected if for every partition of its vertices into two non-empty, disjoint, sets some chordless path with three edges contains vertices from both sets in the partition. As it turns out, p-connectedness generalizes the usual connectedness of graphs and leads, in a natural way, to a unique tree representation for arbitrary graphs.
This paper reviews old and new results, both structural and algorithmic, about p-connectedness along with applications to various graph decompositions