A class of graphs with depression three

An edge ordering   of a graph GG is an injection f:E→Rf:E→R, the set of real numbers. A path in GG for which the edge ordering ff increases along its edge sequence is called an ff-ascent  ; an ff-ascent is maximal   if it is not contained in a longer ff-ascent. The depression   of GG is the smallest integer kk such that any edge ordering ff has a maximal ff-ascent of length at most kk. We characterize the class of graphs with depression three and without adjacent vertices of degree three or higher.