  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.
