Extending Biconnected Straight-Line Planar Drawings

2026-07-28

AI Summary

This study addresses the Partial Drawing Extensibility (PDE) problem for planar graphs when the initial partial drawing is biconnected. By leveraging graph embedding analysis, complexity reductions, and the Existential Theory of the Reals (ETR), the authors establish for the first time that PDE remains NP-hard even when the given subgraph is biconnected and the input graph is subcubic. On the algorithmic side, they devise a polynomial-time algorithm running in $O(p^2 n)$ time under a fixed embedding and develop a fixed-parameter tractable (FPT) algorithm parameterized by the vertex cover number. This work delineates the computational complexity landscape of PDE in the biconnected setting and achieves efficient solutions under fixed embeddings and path-extension constraints.

Abstract

The Partial Drawing Extensibility problem, for short PDE, takes as input a triple $\langle G,H,Γ_H\rangle$, where $G$ is a planar graph, $H$ is a subgraph of $G$, and $Γ_H$ is a straight-line planar drawing of $H$, and asks whether $Γ_H$ can be extended to a straight-line planar drawing of $G$. Patrignani [Int. J. Found. Comput. Sci. (2006)] proved that the PDE problem is NP-hard, exploiting instances in which $H$ is highly disconnected. In this paper, we study the PDE problem under the requirement that the initial partial drawing $Γ_H$ is biconnected. We show that PDE remains NP-hard even for instances in which $H$ is a biconnected graph with faces of bounded size, $G$ is subcubic, and the part of $G$ that is not in $H$ consists of length-$2$ paths. The complexity of PDE remains however open when $H$ is connected (or even biconnected) if $G$ has a fixed embedding. In this setting both a polynomial-time algorithm or an NP-hardness proof seem to be elusive targets. As a step towards tackling this problem, we study instances of PDE in which $H$ is biconnected, $G$ has a fixed embedding, and the rest of the graph consists of $p$ length-2 paths, and present an $O(p^2 n)$-time algorithm, a result in sharp contrast with the NP-hardness of the variable embedding setting. Moreover, with an approach based on the Existential Theory of the Reals, we show that, if $H$ is biconnected, the problem is FPT parameterized by the vertex cover number of $G$, both in a fixed and in a variable embedding setting.

Problem

• Partial Drawing Extensibility
• Biconnected
• Planar Drawing
• Fixed Embedding
• NP-hardness

Innovation

• Partial Drawing Extensibility
• Biconnected Graph
• Fixed Embedding
• Existential Theory of the Reals
• Fixed-Parameter Tractability