Strip Planarity Testing of Embedded Planar Graphs

TL;DR

This paper introduces the strip planarity testing problem, which involves determining if a planar graph can be drawn with edges monotone in the y-direction respecting a given vertex order, and shows it is polynomial-time solvable for fixed embeddings.

Contribution

The paper defines the strip planarity testing problem and proves its polynomial-time solvability when the graph's planar embedding is fixed.

Findings

The problem is polynomial-time solvable with a fixed planar embedding.

Strip planarity relates to clustered, upward, and level planarity variants.

The study advances understanding of planarity testing in constrained graph drawings.

Abstract

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph $G(V,E)$ and a function $\gamma:V \rightarrow \{1,2,\dots,k\}$ and asks whether a planar drawing of $G$ exists such that each edge is monotone in the $y$-direction and, for any $u,v\in V$ with $\gamma(u)<\gamma(v)$, it holds $y(u)<y(v)$. The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. We show that the problem is polynomial-time solvable if $G$ has a fixed planar embedding.