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$(X,\mathcal{E})$为一超图。支撑图$Q$是定义在$X$上的图，满足对任意$E\in\mathcal{E}$，由$E$中元素诱导的$Q$子图是连通的。本文研究为主图中连通子图所定义超图构造支撑图的问题。对于图$G=(V,E)$，令$\mathcal{H}$为$G$的连通子图集合。将$G$的顶点划分为两个集合：\emph{终端顶点}$\mathbf{b}(V)$与\emph{非终端顶点}$\mathbf{r}(V)$。我们在$\mathbf{b}(V)$上定义超图，其中每个$H\in\mathcal{H}$对应一个由$H$中$\mathbf{b}(V)$顶点组成的超边。同时，我们研究为\emph{对偶超图}构造支撑图的问题——该超图定义在$\mathcal{H}$上，每个$v\in \mathbf{b}(V)$对应一个由包含$v$的$\mathcal{H}$中子图组成的超边。事实上，我们为原始问题与对偶问题的共同推广形式——\emph{交超图}构造了支撑图。我们的主要结果表明：若主图$G$具有有界亏格，且$\mathcal{H}$中的子图满足\emph{无交叉}条件，则存在同样具有有界亏格的支撑图。该结果推广了Raman与Ray的研究成果（Rajiv Raman, Saurabh Ray: Constructing Planar Support for Non-Piercing Regions. Discret. Comput. Geom. 64(3): 1098-1122 (2020)）。我们的技术方法为有界亏格曲面上定义超图的覆盖与填装问题提供了统一分析框架。文中还阐述了该结果在超图着色问题中的应用。