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**定义为图 $Q$（顶点集为 $X$），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 $G=(V,E)$，映射 $c:V\to\{\mathbf{r},\mathbf{b}\}$，以及 $G$ 的连通子图族 $\mathcal{H}$，**原始支撑**是顶点集 $\mathbf{b}(V)$ 上的图 $Q$，使得对每个 $H\in \mathcal{H}$，顶点集 $\mathbf{b}(H)=H\cap c^{-1}(\mathbf{b})$ 上的诱导子图 $Q[\mathbf{b}(H)]$ 连通。**对偶支撑**是顶点集 $\mathcal{H}$ 上的图 $Q^*$，使得对每个 $v\in X$，诱导子图 $Q^*[\mathcal{H}_v]$ 连通（其中 $\mathcal{H}_v=\{H\in\mathcal{H}: v\in H\}$）。我们给出宿主图与超边的充分条件，使得所得支撑属于受限图族。主要研究两类图：(1) 若宿主图亏格为 $g$ 且超图满足拓扑条件——**交叉自由**，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 $t$ 且超边满足组合条件——**非穿透性**，则存在树宽为 $O(2^t)$ 的原始与对偶支撑，并证明该指数爆炸在某些情况下不可避免。作为中间情形，我们亦研究宿主图为外平面图的情况。最后展示结果在几何超图的包装覆盖与着色问题中的应用。