AI 中文总结
该研究确定秩宽至多为二的图类的排除顶点子式和排除枢轴子式，通过反转素图单顶点约化定理，利用规范键确定等价类，借助计算机辅助计算得出具体数量，完善了相关列表。
AI 中文摘要
我们确定了秩宽至多为二的图类的排除顶点子式和排除枢轴子式。在局部等价和图同构的意义下，有25个排除顶点子式，分别为8个顶点的1个图、9个顶点的18个图和10个顶点的6个图；有609个排除枢轴子式，分别为8个顶点的2个图、9个顶点的447个图、10个顶点的146个图、11个顶点的10个图和12个顶点的4个图。11 - 16个顶点不存在排除顶点子式，13 - 16个顶点不存在排除枢轴子式。证明是计算机辅助的，通过反转素图的单顶点约化定理，利用各顶点数的素图并扩展一个顶点，借助从相关各向同性系统得到的精确规范键来确定局部等价类，同一键的受限版本对枢轴等价进行分类，顶点子式和枢轴子式计算在16顶点的最终层分别检查了超过\(9.0\times 10^{10}\)和\(4.9\times 10^{11}\)个素扩展。
英文摘要
We determine both the excluded vertex-minors and the excluded pivot-minors for the class of graphs of rank-width at most two. Up to local equivalence and graph isomorphism, there are exactly 25 excluded vertex-minors: 1 graph on 8 vertices, 18 on 9 vertices, and 6 on 10 vertices. Up to pivot equivalence and graph isomorphism, there are exactly 609 excluded pivot-minors: 2 on 8 vertices, 447 on 9 vertices, 146 on 10 vertices, 10 on 11 vertices, and 4 on 12 vertices. No excluded vertex-minor occurs on 11--16 vertices, and no excluded pivot-minor occurs on 13--16 vertices; the author's 16-vertex bound makes both lists complete. The proof is computer-assisted. Instead of enumerating all graphs, we reverse the one-vertex reduction theorem for prime graphs. For each $n$, we retain exactly the prime $n$-vertex graphs of rank-width at most two, modulo local equivalence and isomorphism, and extend those graphs by one vertex. Local-equivalence classes are identified by an exact canonical key obtained from the associated isotropic system, the binary row space of $[I\mid A(G)]$. A restricted version of the same key classifies pivot equivalence exactly. The vertex-minor and pivot-minor computations examine, respectively, more than $9.0\times 10^{10}$ and $4.9\times 10^{11}$ prime extensions in their 16-vertex final layers.
Comments27 pages, 3 figures. Now mention that the graph lists and verification software are available in the public repository https://github.com/sangiloum/rwd2
