Let $\mathcal{H}=(X,\mathcal{E})$ 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. In this paper, we consider hypergraphs defined on a host graph. Given a graph $G=(V,E)$, with $c:V\to\{\mathbf{r},\mathbf{b}\}$, and a collection of connected subgraphs $\mathcal{H}$ of $G$, a primal support is a graph $Q$ on $\mathbf{b}(V)$ such that for each $H\in \mathcal{H}$, the induced subgraph $Q[\mathbf{b}(H)]$ on vertices $\mathbf{b}(H)=H\cap c^{-1}(\mathbf{b})$ is connected. A \emph{dual support} is a graph $Q^*$ on $\mathcal{H}$ s.t. for each $v\in X$, the induced subgraph $Q^*[\mathcal{H}_v]$ is connected, where $\mathcal{H}_v=\{H\in\mathcal{H}: v\in H\}$. We present sufficient conditions on the host graph and hyperedges so that the resulting support comes from a restricted family. We primarily study two classes of graphs: $(1)$ If the host graph has genus $g$ and the hypergraphs satisfy a topological condition of being \emph{cross-free}, then there is a primal and a dual support of genus at most $g$. $(2)$ If the host graph has treewidth $t$ and the hyperedges satisfy a combinatorial condition of being \emph{non-piercing}, then there exist primal and dual supports of treewidth $O(2^t)$. We show that this exponential blow-up is sometimes necessary. As an intermediate case, we also study the case when the host graph is outerplanar. Finally, we show applications of our results to packing and covering, and coloring problems on geometric hypergraphs.

翻译：设 $\mathcal{H}=(X,\mathcal{E})$ 为一个超图。**支撑**是指定义在 $X$ 上的图 $Q$，使得对每个 $E\in\mathcal{E}$，$Q$ 在 $E$ 中元素上诱导的子图是连通的。本文考虑定义在宿主图上的超图。给定图 $G=(V,E)$ 及其顶点染色 $c:V\to\{\mathbf{r},\mathbf{b}\}$，以及 $G$ 的连通子图族 $\mathcal{H}$，**原始支撑**是定义在 $\mathbf{b}(V)$ 上的图 $Q$，使得对每个 $H\in\mathcal{H}$，$Q$ 在顶点集 $\mathbf{b}(H)=H\cap c^{-1}(\mathbf{b})$ 上诱导的子图 $Q[\mathbf{b}(H)]$ 是连通的。**对偶支撑**是定义在 $\mathcal{H}$ 上的图 $Q^*$，使得对每个 $v\in X$，$Q^*$ 在 $\mathcal{H}_v=\{H\in\mathcal{H}: v\in H\}$ 上诱导的子图 $Q^*[\mathcal{H}_v]$ 是连通的。我们给出宿主图与超边的充分条件，使得所得支撑来自受限图族。我们主要研究两类图：（1）若宿主图亏格为 $g$ 且超图满足称为 **交叉自由** 的拓扑条件，则存在亏格至多为 $g$ 的原始支撑与对偶支撑。（2）若宿主图树宽为 $t$ 且超边满足称为 **非穿刺** 的组合条件，则存在树宽为 $O(2^t)$ 的原始支撑与对偶支撑。我们证明这种指数级爆炸有时是不可避免的。作为中间情形，我们还研究了宿主图为外平面图的情况。最后，我们展示结果在几何超图的打包与覆盖问题以及染色问题中的应用。