The depression of a graph and k-kernels
An edge ordering of a graph G is an injection f : E(G) → 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. A k-kernel of a graph G is a set of vertices U ⊆ V (G) such that for any edge ordering f of G there exists a maximal f-ascent of...