诱导荫度和弱诱导荫度
Induced and Weak Induced Arboricities
摘要：
定义图$G$的诱导荫度，记为${\rm ia}(G)$，即使$G$中的$k$诱导森林覆盖$G$的边的最小$k$.这个概念推广了经典的荫度和强色指数概念。对于图的类$\mathcal{F}$和图参数$p$，设$p(\mathcal{F}) = \sup\{p(G) \mid G\in \mathcal{F}\}$。我们证明了${\rm ia}(\mathcal{F})$是有界的，从上面的一个绝对常数只取决于$\mathcal{F}$，即${\rm ia}(\mathcal{F})\neq\infty$当且仅当$χ(\mathcal{F} \nabla \frac{1}{2}) \neq\infty$，其中$\mathcal{F} \nabla \frac{1}{2}$是类的$\frac{1}{2}$ -浅子图从$\mathcal{F}$和$χ$是色数。进一步，当$\mathcal{F}$是平面图类、$d$ -退化图类或树宽至多为$d$的图类时，我们给出了${\rm ia}(\mathcal{F})$的界.具体地说，我们表明，如果$\mathcal{F}$是平面图类，那么$8 \leq {\rm ia}(\mathcal{F}) \leq 10$。此外，我们建立了类似的结果，所谓的弱诱导荫度和星荫度的类图。
英文摘要：
We define the induced arboricity of a graph $G$, denoted by ${\rm ia}(G)$, as the smallest $k$ such that the edges of $G$ can be covered with $k$ induced forests in $G$. This notion generalizes the classical notions of the arboricity and strong chromatic index. For a class $\mathcal{F}$ of graphs and a graph parameter $p$, let $p(\mathcal{F}) = \sup\{p(G) \mid G\in \mathcal{F}\}$. We show that ${\rm ia}(\mathcal{F})$ is bounded from above by an absolute constant depending only on $\mathcal{F}$, that is ${\rm ia}(\mathcal{F})\neq\infty$ if and only if $χ(\mathcal{F} \nabla \frac{1}{2}) \neq\infty$, where $\mathcal{F} \nabla \frac{1}{2}$ is the class of $\frac{1}{2}$-shallow minors of graphs from $\mathcal{F}$ and $χ$ is the chromatic number. Further, we give bounds on ${\rm ia}(\mathcal{F})$ when $\mathcal{F}$ is the class of planar graphs, the class of $d$-degenerate graphs, or the class of graphs having tree-width at most $d$. Specifically, we show that if $\mathcal{F}$ is the class of planar graphs, then $8 \leq {\rm ia}(\mathcal{F}) \leq 10$. In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.
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