[论文解读] The Jordan-Brouwer theorem for graphs

Oliver Knill|arXiv (Cornell University)|Jun 22, 2015

Digital Image Processing Techniques参考文献 39被引用 9

一句话总结

本文在图论中建立了乔丹-布劳尔-舒恩费尔斯定理的离散类比，证明了嵌入在d-球图中的(d−1)-球面将该图分割为两个互补的d-球。通过基于Evako的可缩性及球面/球体概念的归纳图定义，证明利用同伦形变和交点数论证，表明互补分量为可缩的d-球，将经典拓扑学推广至有限图。

ABSTRACT

We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic definitions are due to Evako: the unit sphere of a vertex x of a graph G=(V,E) is the graph generated by {y | (x,y) in E} Inductively, a finite simple graph is called contractible if there is a vertex x such that both its unit sphere S(x) as well as the graph generated by V-{x} are contractible. Inductively, still following Evako, a d-sphere is a finite simple graph for which every unit sphere is a (d-1)-sphere and such that removing a single vertex renders the graph contractible. A d-ball B is a contractible graph for which each unit sphere S(x) is either a (d-1)-sphere in which case x is called an interior point, or S(x) is a (d-1)-ball in which case x is called a boundary point and such that the set of boundary point vertices generates a (d-1)-sphere. These inductive definitions are based on the assumption that the empty graph is the unique (-1)-sphere and that the one-point graph K_1 is the unique 0-ball and that K_1 is contractible. The theorem needs the following notion of embedding: a sphere H is embedded in a graph G if it is a sub-graph of G and if any intersection with any finite set of mutually adjacent unit spheres is a sphere. A knot of co-dimension k in G is a (d-k)-sphere H embedded in a d-sphere G.

研究动机与目标

• 通过离散拓扑定义，将经典乔丹-布劳尔-舒恩费尔斯定理推广至有限简单图。
• 基于Evako框架，通过可缩性和单位球面条件，归纳地定义图中的球面和球体。
• 建立在d-球图中嵌入的(d−1)-球面将图分割为两个互补的d-球这一结论。
• 在离散设定中，利用同伦形变和交点数论证，提供一个构造性证明。
• 证明在连续形变下，互补分量保持为d-球，同时保持嵌入结构。

提出的方法

• 采用Evako的归纳定义：d-球是一个图，其中每个单位球面都是(d−1)-球，且移除一个顶点后得到的图是可缩的。
• 将d-球定义为一个可缩图，其边界顶点构成一个(d−1)-球，而内部顶点的单位球面为(d−1)-球。
• 采用一种嵌入概念，即H与任意一组邻近单位球面的交集为一个球面，以确保拓扑正则性。
• 应用同伦形变步骤H → H Δ x，将H替换为与d-单纯形x对称差异的图，以减少一个分量中的顶点数。
• 使用交点数论证：若两个分量相连，则存在一条与H有奇数个交点的闭路径，与交点引理矛盾。
• 依赖于χ(A) + χ(B) = 2 和 χ(G) = 1 + (−1)^d，且χ(H) = 1 − (−1)^d，以支持拓扑一致性。

实验结果

研究问题

• RQ1 能否在有限简单图的上下文中，通过离散拓扑方法重新表述并证明经典乔丹-布劳尔-舒恩费尔斯定理？
• RQ2 在什么条件下，嵌入在d-球图中的(d−1)-球面会将图分割为两个互补分量，且这两个分量均为d-球？
• RQ3 同伦形变如何在离散图中保持球面的嵌入与拓扑结构？
• RQ4 路径与嵌入球面的交点数在证明分离性与连通性方面起什么作用？
• RQ5 是否可以将被球面界定的分量结构简化为一个单纯形，同时保持d-球的性质？

主要发现

• (d−1)-球面H嵌入在d-球图G中，将G分割为两个互补分量A和B，满足A ∪ B = G且A ∩ B = H。
• 分量A和B均为d-球，即它们是可缩图，且其边界为(d−1)-球。
• 证明使用了形变过程H → H Δ x，该过程减少了其中一个分量中的顶点数，最终将H简化为一个单纯形，从而证明该分量为d-球。
• 交点数论证表明，若A和B不连通，则会因一条与H有奇数个交点的闭路径而产生矛盾。
• 即使H的补集无顶点，只要形变过程保持嵌入和拓扑结构，该结论依然成立。
• 该定理在较弱假设下仍成立，即G为单连通图而非必为d-球，但完整结论要求G为d-球。