The role of twins in computing planar supports of hypergraphs

TL;DR

This paper investigates the impact of twin vertices on the existence of planar supports in hypergraphs, providing bounds and algorithms that improve hypergraph drawing and network design methods.

Contribution

It proves that the number of twins needed for a hypergraph to have a planar support depends only on its number of hyperedges and offers a fixed-parameter linear-time algorithm for certain cases.

Findings

Number of twins needed depends only on hyperedge count

Explicit upper bounds for $r$-outerplanar supports

Linear-time algorithm for fixed parameters

Abstract

A support or realization of a hypergraph $H$ is a graph $G$ on the same vertex as $H$ such that for each hyperedge of $H$ it holds that its vertices induce a connected subgraph of $G$. The NP-hard problem of finding a planar support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins -- pairs of vertices that are in precisely the same hyperedges -- can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with $m$ hyperedges to have an $r$-outerplanar support, which depends only on $r$ and $m$. Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing $r$-outerplanar supports for hypergraphs with $m$ hyperedges if $m$ and $r$ are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters $m$ and $r$.