The central theorem of topological graph theory states that the graph minor relation is a well-quasi-order on graphs. It has far-reaching consequences, in particular in the study of graph structures and the design of (parameterized) algorithms. In this article, we study two embedded versions of classical minor relations from structural graph theory and prove that they are also well-quasi-orders on general or restricted classes of embedded planar graphs. These embedded minor relations appear naturally for intrinsically embedded objects, such as knot diagrams and surfaces in $\mathbb{R}^3$.
Handling the extra topological constraints of the embeddings requires careful analysis and extensions of classical methods for the more constrained embedded minor relations. We prove that the embedded version of immersion induces a well-quasi-order on bounded carving-width plane graphs by exhibiting particularly well-structured tree-decompositions and leveraging a classical argument on well-quasi-orders on forests. We deduce that the embedded graph minor relation defines a well-quasi-order on plane graphs via their directed medial graphs, when their branch-width is bounded. We conclude that the embedded graph minor relation is a well-quasi-order on all plane graphs, using classical grids theorems in the unbounded branch-width case.
Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals
给定一个无向的边加权图G和一对顶点（称为终端对），最小多割问题要求一个最小权的边集，使得删除这些边后，每对的两个终端属于图的不同连通分支。依赖于拓扑技术，我们提供了一个多项式时间算法的情况下，G是嵌入在一个固定的曲面上的亏格g（例如，当G是平面时），并且具有固定数量t的端子。运行时间是一个多项式的次数为O（sqrt{g^2+gt}）的输入大小。在平面情况下，我们的结果纠正了Bentz [Int. Workshop on Parameterized and Exact Computation，109-119，2012]的扩展摘要中的错误。最小多割问题也是多路割问题的一个推广，也称为多路割问题。多端切割问题;即使对于这种特殊情况，也没有专门的算法用于嵌入表面上的图形。
