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

09/24/2019

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by   Wojciech Nadara, et al.

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The maximum average degree mad(G) of a graph G is the maximum average degree over all subgraphs of G. In this paper we prove that for every G and positive integer k such that mad(G) > k there exists S ⊆ 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 there exists an independent set I in G such that mad(G-I) <mad(G)-1 and an induced forest F such that mad(G-F) <mad(G) - 2.