On Supports for graphs of bounded genus

TL;DR

The paper advances the theory of sparse representations for hypergraphs defined by connected subgraphs of a host graph, focusing on graphs embedded on surfaces with bounded genus. It introduces cross-free and abab-free concepts and proves that cross-free intersection systems on genus-g hosts admit intersection supports of genus at most g, yielding primal and dual supports as well. Leveraging these supports, the authors derive PTASes for a broad class of packing and covering problems on surfaces and obtain coloring guarantees for geometric hypergraphs, thereby unifying and extending prior plane-based results. The work provides a graph-theoretic framework for hypergraph problems in bounded-genus surfaces, with applications to packing, covering, and coloring, and highlights several open questions about algorithmic construction and complexity.

Abstract

Let $(X,\mathcal{E})$ 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. We consider the problem of constructing a support for hypergraphs defined by connected subgraphs of a host graph. For a graph $G=(V,E)$, let $\mathcal{H}$ be a set of connected subgraphs of $G$. Let the vertices of $G$ be partitioned into two sets the \emph{terminals} $\mathbf{b}(V)$ and the \emph{non-terminals} $\mathbf{r}(V)$. We define a hypergraph on $\mathbf{b}(V)$, where each $H\in\mathcal{H}$ defines a hyperedge consisting of the vertices of $\mathbf{b}(V)$ in $H$. We also consider the problem of constructing a support for the \emph{dual hypergraph} - a hypergraph on $\mathcal{H}$ where each $v\in \mathbf{b}(V)$ defines a hyperedge consisting of the subgraphs in $\mathcal{H}$ containing $v$. In fact, we construct supports for a common generalization of the primal and dual settings called the \emph{intersection hypergraph}. As our main result, we show that if the host graph $G$ has bounded genus and the subgraphs in $\mathcal{H}$ satisfy a condition of being \emph{cross-free}, then there exists a support that also has bounded genus. Our results are a generalization of the results of Raman and Ray (Rajiv Raman, Saurabh Ray: Constructing Planar Support for Non-Piercing Regions. Discret. Comput. Geom. 64(3): 1098-1122 (2020)). Our techniques imply a unified analysis for packing and covering problems for hypergraphs defined on surfaces of bounded genus. We also describe applications of our results for hypergraph colorings.

Figures (7)

• Figure 1: Support for hypergraph defined by disks and points in the plane.
• Figure 2: (a) and (b): Primal and Dual hypergraphs on the graph system $(G,\mathcal{H})$ where $G$ is the graph on vertices $\{a,b\ldots,f\}$, and $\mathcal{H}=\{H_1,\ldots H_4\}$ with $H_1=\{a,b,c,d\}$, $H_2=\{c,d,e\}$, $H_3=\{a,b,f,e\}$, $H_4=\{a,b,c,e\}$; (c): Intersection hypergraph for $(G,\mathcal{H}',\mathcal{K})$ with $\mathcal{H}'=\{H'_1,\ldots,H'_4\}$ and $\mathcal{K}=\{K_1,\ldots,K_3\}$ where $H'_1=\{a,b,c\}$, $H'_2=\{a,b,c,e\}$, $H'_3=\{c,e,f\}$, $H'_4=\{b,e,d\}$, and $K_1=\{c,d\}$, $K_2=\{b,f\}$, $K_3=\{e,d\}$. Figures in (d)-(f) show the respective support graphs.
• Figure 3: Vertex Bypassing
• Figure 4: Finding a non-blocking chord to join two disjoint runs of $K_0$.
• Figure 5: An arrangement of Pseudodisks (left), Piercing (middle), and Non-piercing (right) regions in the plane.

Theorems & Definitions (79)

• Definition 1: Embedding of a graph
• Definition 2: Genus
• Theorem 3: gilbert1984separator
• Definition 4: Reduced graph
• Definition 5: Cross-free at $v$
• Definition 6: Non-piercing subgraphs
• Theorem 6
• Theorem 6
• Theorem 6
• Theorem 6