The role of twins in computing planar supports of hypergraphs

Abstract

A support or realization of a hypergraph $\mathcal{H}$ is a graph \(G\) on the same vertex set as \(\mathcal{H}\) such that for each hyperedge of $\mathcal{H}$ 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 $r$.