小网格EPG表示
EPG-representations with small grid-size
摘要：
在图$G$的EPG表示中，每个顶点由矩形网格中的路径表示，并且当且仅当表示$v$和$w$的路径共享网格边时，$(v,w)$是$G$中的边。要求表示边的路径是x-单调的，或者更强的，x-和y-单调的，产生了EPG表示的三个自然变体，一个是边没有单调性要求的，另一个是具有上述单调性要求的。本文的重点是了解如何小的网格可以实现这样的EPG表示相对于各种图形参数。我们发现，有$m$边缘图，需要一个网格的区域$Ω(m)$在任何变体的EPG表示。类似地，在EPG表示的任何变体中，存在需要高度$Ω(k)$和面积$Ω(kn)$的路径宽度- $k$图。我们证明了一个匹配上界的$O(kn)$面积的所有路径宽度$k$图中的最强模型，其中的边缘都需要是x-和y-单调。因此，在这个最强的模型中，结果意味着，例如，$O(n)$，$O(n \log n)$和$O(n^{3/2})$面积边界分别为有界路径宽度图，有界树宽度图和所有类别的图，排除一个固定的子。对于边的单调性没有限制的模型，对于某些图类，例如有界树宽图的$O(n)$面积界和有界亏格图的$O(n \log^2 n)$界，可以得到更强的结果.
英文摘要：
In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are $m$-edge graphs that require a grid of area $Ω(m)$ in any variant of EPG-representations. Similarly there are pathwidth-$k$ graphs that require height $Ω(k)$ and area $Ω(kn)$ in any variant of EPG-representations. We prove a matching upper bound of $O(kn)$ area for all pathwidth-$k$ graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, $O(n)$, $O(n \log n)$ and $O(n^{3/2})$ area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an $O(n)$ area bound for bounded treewidth graphs and $O(n \log^2 n)$ bound for graphs of bounded genus.
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