Let $(X,\mathcal{E})$ be a hypergraph. A support is a graph $Q$ on $X$ such that for each $E\in\mathcal{E}$, the subgraph of $Q$ induced on the elements in $E$ is connected. We consider the problem of constructing a support for hypergraphs defined by connected subgraphs of a host graph. For a graph $G=(V,E)$, let $\mathcal{H}$ be a set of connected subgraphs of $G$. Let the vertices of $G$ be partitioned into two sets the \emph{terminals} $\mathbf{b}(V)$ and the \emph{non-terminals} $\mathbf{r}(V)$. We define a hypergraph on $\mathbf{b}(V)$, where each $H\in\mathcal{H}$ defines a hyperedge consisting of the vertices of $\mathbf{b}(V)$ in $H$. We also consider the problem of constructing a support for the \emph{dual hypergraph} - a hypergraph on $\mathcal{H}$ where each $v\in \mathbf{b}(V)$ defines a hyperedge consisting of the subgraphs in $\mathcal{H}$ containing $v$. In fact, we construct supports for a common generalization of the primal and dual settings called the \emph{intersection hypergraph}. As our main result, we show that if the host graph $G$ has bounded genus and the subgraphs in $\mathcal{H}$ satisfy a condition of being \emph{cross-free}, then there exists a support that also has bounded genus. Our results are a generalization of the results of Raman and Ray (Rajiv Raman, Saurabh Ray: Constructing Planar Support for Non-Piercing Regions. Discret. Comput. Geom. 64(3): 1098-1122 (2020)). Our techniques imply a unified analysis for packing and covering problems for hypergraphs defined on surfaces of bounded genus. We also describe applications of our results for hypergraph colorings.

翻译：设$(X,\mathcal{E})$为一个超图。支撑是指图$Q$（顶点集为$X$），使得对每个$E\in\mathcal{E}$，$Q$中由$E$中元素诱导的子图是连通的。我们考虑为宿主图中连通子图所定义的超图构造支撑的问题。对于图$G=(V,E)$，设$\mathcal{H}$为$G$的一组连通子图。将$G$的顶点划分为两类：\emph{终端}顶点$\mathbf{b}(V)$和\emph{非终端}顶点$\mathbf{r}(V)$。我们在$\mathbf{b}(V)$上定义一个超图，其中每个$H\in\mathcal{H}$定义一个超边，该超边由$H$中属于$\mathbf{b}(V)$的顶点组成。我们还考虑了为\emph{对偶超图}构造支撑的问题——该超图以$\mathcal{H}$为顶点集，其中每个$v\in \mathbf{b}(V)$定义一个超边，该超边由$\mathcal{H}$中包含$v$的子图组成。实际上，我们为原始设定和对偶设定的共同推广——称为\emph{交超图}——构造了支撑。作为主要结果，我们证明：如果宿主图$G$具有有界亏格，且$\mathcal{H}$中的子图满足\emph{交叉自由}条件，则存在一个同样具有有界亏格的支撑。我们的结果推广了Raman与Ray的研究（Rajiv Raman, Saurabh Ray: Constructing Planar Support for Non-Piercing Regions. Discret. Comput. Geom. 64(3): 1098-1122 (2020)）。本文方法为有界亏格曲面上定义的超图的覆盖与包装问题提供了统一分析框架。我们还描述了结果在超图染色中的应用。