On Supports for graphs of bounded genus

2025-03-27

AI Summary

This paper addresses the problem of constructing low-genus support graphs for hypergraphs defined by connected subgraphs (and their duals and intersection variants) on bounded-genus graphs. Specifically, given a host graph embedded on a surface of genus $g$, we establish—for the first time—that if the hyperedges correspond to a crossing-free family of connected subgraphs, then the intersection hypergraph admits a support graph whose genus depends only on $g$. Our method integrates combinatorial topology, graph embedding theory, and structural analysis of crossing-free configurations, thereby generalizing prior results restricted to planar surfaces. The main contribution is the first unified existence theorem for support graphs on arbitrary bounded-genus surfaces, providing a general topological framework for combinatorial optimization problems on such hypergraphs—including coloring, packing, and covering.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Let $(X,mathcal{E})$ 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. 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 $Hinmathcal{H}$ 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 $vin mathbf{b}(V)$ 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.

Problem

• Constructing supports for hypergraphs with bounded genus
• Analyzing intersection hypergraphs in primal and dual settings
• Extending planar support results to non-piercing regions

Innovation

• Constructs support for hypergraphs on bounded genus graphs
• Generalizes primal and dual settings via intersection hypergraph
• Unifies analysis for packing and covering problems