We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(log n + log g(log log g)3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log4g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for low-genus graphs, and to find in linear time a tree-decomposition of low-genus low-diameter graphs.
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机译:我们提供了一种数据结构,用于在分摊时间 O I>(每次在属的表面上更新log n I> + log g I>(log log g I>) 3 SUP>) g I>;我们还可以同时测试表面的方向性,并在 O I>(log n I> + log 4 SUP> g I>)。我们的数据结构允许边缘插入和删除以及双重操作;这些操作可能会隐式更改嵌入表面的属。我们应用类似的思想来改进低类图的分离定理中的常数因子,并在线性时间内找到低类低直径图的树分解。
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