Well-Formed Separator Sequences, with an Application to Hypergraph Drawing

TL;DR

This paper proves that the Planar Support problem for hypergraphs is fixed-parameter tractable by developing new structural separator sequences for certain planar graphs, enabling effective data reduction.

Contribution

It introduces novel structural results for r-outerplanar triangulated disks and applies them to establish fixed-parameter tractability of the Planar Support problem.

Findings

Planar Support is fixed-parameter tractable with respect to hyperedges and outerplanarity.

Structural separator sequences enable data reduction for specific planar graphs.

A problem kernel for Planar Support is obtained through these structural insights.

Abstract

Given a hypergraph $H$, the Planar Support problem asks whether there is a planar graph $G$ on the same vertex set as $H$ such that each hyperedge induces a connected subgraph of $G$. Planar Support is motivated by applications in graph drawing and data visualization. We show that Planar Support is fixed-parameter tractable when parameterized by the number of hyperedges in the input hypergraph and the outerplanarity number of the sought planar graph. To this end, we develop novel structural results for $r$-outerplanar triangulated disks, showing that they admit sequences of separators with structural properties enabling data reduction. This allows us to obtain a problem kernel for Planar Support, thus showing its fixed-parameter tractability.