对平面性参数扩展的分支宽度近似
Approximating branchwidth on parametric extensions of planarity
AI总结：
本文提出了一种对平面性参数扩展图的分支宽度近似算法，通过构造特定子图实现常数加性近似。
摘要：
分支宽度是罗伯逊和塞伊姆引入的一种度量，用于衡量图的树分解性，作为树宽度的替代。分支宽度可以在平面图上通过塞伊姆和托马斯的著名“Ratcatcher”算法进行多项式计算。我们探讨如何将该算法扩展到超出平面图的闭包图类，如下：令H₁为可嵌入在扭棱柱中的图，H₂为可嵌入在项目平面上的图。我们证明，对于每一个{H₁,H₂}-无小图G，存在一个子图G'，其分支宽度与G的分支宽度相差一个仅依赖于H₁和H₂的常数。此外，G'允许一个树分解，其中所有扭结都是平面的。这种分解允许分支宽度的常数加性近似：对于{H₁,H₂}-无小图，存在一个常数c（依赖于H₁和H₂）和一个O(|V(G)|³)时间的算法，该算法在给定图G时，输出一个值b，使得G的分支宽度在b和b+c之间。
英文摘要：
The branchwidth of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher'' algorithm of Seymour and Thomas. We explore how this algorithm can be extended to minor-closed graph classes beyond planar graphs, as follows: Let $H_{1}$ be a graph embeddable in the torus and $H_{2}$ be a graph embeddable in the projective plane. We prove that every $\{H_{1},H_{2}\}$-minor free graph $G$ contains a subgraph $G'$ whose branchwidth differs from that of $G$ by a constant depending only on $H_1$ and $H_2$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition allows for a constant-additive approximation of branchwidth: For $\{H_{1},H_{2}\}$-minor free graphs, there is a constant $c$ (depending on $H_{1}$ and $H_{2}$) and an $\mathcal{O}(|V(G)|^{3})$-time algorithm that, given a graph $G$, outputs a value $b$ such that the branchwidth of $G$ is between $b$ and $b+c$.
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