A class of graphs with depression three

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