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

TL;DR

This paper proves that for any graph with a certain maximum average degree, one can find a subset of vertices whose removal significantly reduces the degree, and this subset can be efficiently computed.

Contribution

It introduces a method to decrease the maximum average degree of a graph by removing a specific subgraph or independent set, with polynomial-time algorithms.

Findings

Existence of a subset reducing mad by at least k

Efficient polynomial-time computation of such subsets

Application to independent sets and forests

Abstract

The maximum average degree $\mathrm{mad}(G)$ 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 $\mathrm{mad}(G) \ge k$ there exists $S \subseteq V(G)$ such that $\mathrm{mad}(G - S) \le \mathrm{mad}(G) - k$ 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 $\mathrm{mad}(G-I) \le \mathrm{mad}(G)-1$ and an induced forest $F$ such that $\mathrm{mad}(G-F) \le \mathrm{mad}(G) - 2$.