Decreasing the maximum average degree by deleting an independent set or a d-degenerate subgraph

The maximum average degree mad(G) of a graph G is the maximum over all subgraphs of G, of the average degree of the subgraph. In this paper, we prove that for every G and positive integer k such that mad(G) >= k there exists S subset of V (G) such that mad(G - S) <= mad(G) - k and G[S] is (k - 1)-degenerate. Moreover, such S can be computed in polynomial time. In particular, if G contains at least one edge then there exists an independent set I in G such that mad(G - I) <= mad(G) - 1 and if G contains a cycle then there exists an induced forest F such that mad(G - F) <= mad(G) - 2. As a side result, we also obtain a subexponential bound on the diameter of reconfiguration graphs of generalized colourings of graphs with bounded value of their mad.