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On Hypergraph Supports

Rajiv Raman, Karamjeet Singh

Abstract

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.

On Hypergraph Supports

Abstract

Let be a hypergraph. A support is a graph on such that for each , the subgraph of induced on the elements in is connected. In this paper, we consider hypergraphs defined on a host graph. Given a graph , with , and a collection of connected subgraphs of , a primal support is a graph on such that for each , the induced subgraph on vertices is connected. A \emph{dual support} is a graph on s.t. for each , the induced subgraph is connected, where . 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: If the host graph has genus 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 . If the host graph has treewidth and the hyperedges satisfy a combinatorial condition of being \emph{non-piercing}, then there exist primal and dual supports of treewidth . 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.
Paper Structure (23 sections, 35 theorems, 1 equation, 8 figures)

This paper contains 23 sections, 35 theorems, 1 equation, 8 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Graph classes
  5. Contribution
  6. Non-piercing and Cross-free Systems
  7. Bounded Genus Graphs
  8. Vertex Bypassing
  9. Non-blocking chords in $abab$-free hypergraphs
  10. Construction of Supports
  11. Primal Support
  12. Dual Support
  13. Intersection Support
  14. Outerplanar Graphs
  15. Graphs of Bounded Treewidth
  16. ...and 8 more sections

Key Result

Proposition 6

Let $(G,\mathcal{H})$ be a graph system. Let $\mathcal{H}'\subseteq\mathcal{H}$ be maximal such that $\forall\; H, H'\in\mathcal{H}'$, $H\setminus H'\neq\emptyset$. Let $Q'$ be a dual support for $(G,\mathcal{H}')$. Then, there is a dual support $Q^*$ for $(G,\mathcal{H})$ such that

Figures (8)

  • Figure 1: Support for hypergraph defined by disks and points in the plane.
  • Figure 2: (a) and (c): Primal and Dual hypergraphs both having hyperedges $H_1=\{a,b,c,d\}$, $H_2=\{c,d,e\}$, $H_3=\{a,b,f,e\}$, $H_4=\{a,b,c,e\}$; (e): Intersection hypergraph with hyperedges $H_1=\{a,b,c\}$, $H_2=\{a,b,e\}$, $H_3=\{c,e,f\}$, $H_4=\{b,e,d\}$, and $K_1=\{c,d\}$, $K_2=\{b,f\}$, $K_3=\{e,d\}$.
  • Figure 3: Finding a non-blocking chord to join two disjoint runs of $K_0$.
  • Figure 4:
  • Figure 5: Primal and dual problems with piercing subgraphs that do not possess a primal or dual support of bounded treewidth.
  • ...and 3 more figures

Theorems & Definitions (77)

  • Definition 1: Reduced graph
  • Definition 2: Cross-free at $v$
  • Definition 3: Non-piercing
  • Definition 4: Embedding of a graph
  • Definition 5: Genus
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • Definition 8: VB$(v)$
  • ...and 67 more