The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that $|E(G)|\ge α|V(G)|^2$ for some fixed $α>0$. While a constant factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph $G$ of order $n$ and given $\varepsilon>0$, returns an integer $g$ such that $G$ has an embedding into a surface of genus $g$, and this is $\varepsilon$-close to a minimum genus embedding in the sense that the minimum genus $\mathsf{g}(G)$ of $G$ satisfies: $\mathsf{g}(G)\le g\le (1+\varepsilon)\mathsf{g}(G)$. The running time of the algorithm is $O(f(\varepsilon)\,n^2)$, where $f(\cdot)$ is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is $g$. This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time $O(f_1(\varepsilon)\,n^2)$.
Las Vergnas多项式是Tutte多项式到胞腔嵌入图的扩展。它是由Michel Las Vergnas在1978年引入的，作为拟阵态射的Tutte多项式的特例。虽然一般的Tutte多项式的态射拟阵有一套完整的删除-收缩关系，其专门细胞嵌入图没有。在这里，我们扩展Las Vergnas多项式的伪曲面图。我们表明，在这种设置中，我们可以定义删除和收缩嵌入图一致的删除和收缩的基本拟阵的角度来看，从而产生一个版本的拉斯维加斯Vergnas多项式完整的递归定义。这也使我们能够更深入地了解Las Vergnas多项式，Bollobas-Riordan多项式和Krushkal多项式之间的关系。我们也借此机会扩展一些拉斯维加斯Vergnas的结果欧拉电路从图表面的低亏格表面的任意亏格。
The stable set problem in graphs with bounded genus and bounded odd cycle packing number
考虑不含$ k $节点不相交奇圈的图族，其中$ k $是常数。确定这种图的稳定集问题的复杂性$ G $是一个长期存在的问题。对于$ G $可进一步嵌入有界亏格曲面（可能不可定向）的情形，我们给出了一个多项式时间算法.此外，我们得到多项式大小的扩展配方各自的稳定集多面体。为此，我们证明了嵌入固定曲面的图的$2$边奇圈满足Erdens-Pósa性质.这扩展了奇圈满足嵌入固定可定向曲面的图中的Erdens-Pósa性质的事实（Kawarabayashi & Nakamoto，2007）。最终，我们的研究结果使我们能够减少原来的问题，找到一个最小成本的非负整数循环的某个同源类，这在我们的情况下是有效的解决方案的问题。
