本地和联盟Boxicity
Local and Union Boxicity
摘要：
图$H$的盒度$\operatorname{box}(H)$是最小整数$d$，使得$H$是$d$区间图的交集，或者等价地，$H$是$\mathbb{R}^d$中轴对齐盒的交图。这些交集表示可以被解释为覆盖$H$的补图$H^c$与余区间图的表示，即区间图的补图。我们遵循最近的全局，局部和折叠覆盖数框架（Knauer和Ueckerdt，离散数学339（2016））来定义两个新参数：局部盒性$\operatorname{box}_\ell(H)$和$H$的并盒性$\overline{\operatorname{box}}(H)$。$H$的并盒度是最小的$d$，使得$H^c$可以被余区间图的点不交并$d$覆盖，而$H$的局部盒度是最小的$d$，使得$H^c$可以被余区间图覆盖，最多在每个顶点都有$d$.我们表明，对于每一个图$H$，我们有$\operatorname{box}_\ell(H) \leq \overline{\operatorname{box}}(H) \leq \operatorname{box}(H)$，这些不等式中的每一个可以是任意远。此外，我们还表明，当地和工会boxicity的特点是相交表示适当的轴对齐的盒子在$\mathbb{R}^d$。我们证明了一些引人注目的例子，在某种意义上，局部盒性是一个更好的指示图的复杂性，比经典的盒性。
英文摘要：
The boxicity $\operatorname{box}(H)$ of a graph $H$ is the smallest integer $d$ such that $H$ is the intersection of $d$ interval graphs, or equivalently, that $H$ is the intersection graph of axis-aligned boxes in $\mathbb{R}^d$. These intersection representations can be interpreted as covering representations of the complement $H^c$ of $H$ with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of global, local and folded covering numbers (Knauer and Ueckerdt, Discrete Mathematics 339 (2016)) to define two new parameters: the local boxicity $\operatorname{box}_\ell(H)$ and the union boxicity $\overline{\operatorname{box}}(H)$ of $H$. The union boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with $d$ vertex-disjoint unions of co-interval graphs, while the local boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with co-interval graphs, at most $d$ at every vertex. We show that for every graph $H$ we have $\operatorname{box}_\ell(H) \leq \overline{\operatorname{box}}(H) \leq \operatorname{box}(H)$ and that each of these inequalities can be arbitrarily far apart. Moreover, we show that local and union boxicity are also characterized by intersection representations of appropriate axis-aligned boxes in $\mathbb{R}^d$. We demonstrate with a few striking examples, that in a sense, the local boxicity is a better indication for the complexity of a graph, than the classical boxicity.
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