AI 中文总结
该研究针对平面图的子集（简单外平面图与简单近外平面图），明确给出其 scramble 数的边界，解答了任意平面图的 scramble 数是否遵循 $O(\sqrt{n})$ 这一开放问题。
AI 中文摘要
对于平面图，已知其树宽（treewidth）受限于 $O(\sqrt{n})$，其中 $n$ 是图的顶点数。与树宽相关的不变量是图的 scramble 数。Connor 等人最近证明，最大度有界的平面图的 scramble 数受限于 $O(\sqrt{n})$。一个开放问题是，任意平面图的 scramble 数是否也遵循这一 bound。我们针对平面图的一个子集——简单外平面图和简单近外平面图，给出了明确的边界，从而给出了确定的答案。
英文摘要
For planar graphs, it is known that their treewidth is bounded by $O(\sqrt{n})$, where $n$ is the number of vertices of the graph. A related invariant to treewidth, is the scramble number of graphs. Recently, Connor et. al proved that planar graphs of bounded maximal degree have scramble number bounded by $O(\sqrt{n})$. An open question is whether the scramble number of any planar graph follows this same bound. We give a definitive answer with an explicit bound for a subset of planar graphs, the simple outerplanar graphs and the simple near outerplanar graphs.
Comments13 pages, 3 figures
