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- W4361806609 abstract "Let $mathcal{H}=(X,mathcal{E})$ be a hypergraph. A support is a graph $Q$ on $X$ such that for each $Einmathcal{E}$, 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 $c:Vto{mathbf{r},mathbf{b}}$, 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 $Hin mathcal{H}$, the induced subgraph $Q[mathbf{b}(H)]$ on vertices $mathbf{b}(H)=Hcap c^{-1}(mathbf{b})$ is connected. A emph{dual support} is a graph $Q^*$ on $mathcal{H}$ s.t. for each $vin X$, the induced subgraph $Q^*[mathcal{H}_v]$ is connected, where $mathcal{H}_v={Hinmathcal{H}: vin H}$. 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." @default.
- W4361806609 created "2023-04-05" @default.
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- W4361806609 date "2023-03-29" @default.
- W4361806609 modified "2023-10-01" @default.
- W4361806609 title "On Hypergraph Supports" @default.
- W4361806609 doi "https://doi.org/10.48550/arxiv.2303.16515" @default.
- W4361806609 hasPublicationYear "2023" @default.
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