AI 中文总结
该研究针对图论中荫度与退化度的结构差距问题，证明k-荫度图可分解为k-退化子图与低度子图，给出分解条件并提出多项式时间算法，同时证明一般图相关判定问题为NP完全。
AI 中文摘要
荫度和退化度是衡量图稀疏性的两个基本且密切相关的图参数。每个k-退化图都是k-荫度图，但部分k-荫度图仅为(2k-1)-退化图。不过，每个含n个顶点的极大k-荫度多重图、每个含n个顶点的极大k-退化多重图，都恰好有k(n-1)条边。这些基本观察引出一个自然的结构问题：k-荫度图与k-退化图的差距有多大？我们通过证明二者差距至多为(k-1)有界度图来回答该问题。更具体地说，我们证明k-荫度多重图存在(k,k-1)-分解，即其边可划分为两个多重集：一个构成k-退化多重图，另一个构成每个顶点度数至多为k-1的多重图。此外，我们对所有可能的此类分解类型给出完整刻画：对任意整数k≥1及d,h≥0，每个k-荫度多重图存在(d,h)-分解当且仅当d≥k且d+h≥2k-1。我们的证明是构造性的，且给出了生成此类分解的多项式时间算法。相比之下，我们证明一般图（无荫度约束）的相关判定问题是NP完全的。
英文摘要
Arboricity and degeneracy are two fundamental and closely related graph parameters that measure the sparsity of a graph. Every $k$-degenerate graph is $k$-arboric, but some $k$-arboric graphs are only $(2k-1)$-degenerate. However, every maximal $k$-arboric multigraph with $n$ vertices and every maximal $k$-degenerate multigraph with $n$ vertices has exactly $k(n-1)$ edges. These basic observations lead to a natural structural question: How far are $k$-arboric graphs from being $k$-degenerate? We answer this question by showing that: By at most a $(k-1)$-bounded-degree graph apart. More specifically, we prove that a $k$-arboric multigraph admits a $(k,k-1)$-decomposition, that is, its edges can be partitioned into two multisets such that one spans a $k$-degenerate multigraph and the other spans a multigraph with every vertex having degree at most $k-1$. Moreover, we provide a complete characterisation of all possible such decomposition types. Namely, for any integers $k \ge 1$ and $d,h \ge 0$ we show that every $k$-arboric multigraph admits a $(d,h)$-decomposition if and only if $d\geq k$ and $d+h\geq 2k-1$. Our proofs are constructive and we present a polynomial time algorithm that produces such decompositions. By contrast, we show that related decision problems for general graphs (without constraints on the arboricity) are NP-complete.
