Realizing Planar Linkages in Polygonal Domains

2026-04-07

AI Summary

This study investigates the computational complexity of realizing non-rigid planar linkage mechanisms within a given polygonal region, with particular emphasis on scenarios where the positions of some vertices are prescribed. By leveraging parameterized complexity theory, graph embedding techniques, and geometric constraint solving, the work establishes the first parameterized complexity results for this problem: it is shown to be in XP and W[1]-hard with respect to the size of the underlying graph, and remains NP-hard even in the absence of rigid components. Furthermore, the paper presents a linear-time algorithm for the special case of three-edge paths with arbitrary edge lengths embedded in a convex polygonal domain.

Abstract

A linkage $\mathcal{L}$ consists of a graph $G=(V,E)$ and an edge-length function $\ell$. Deciding whether $\mathcal{L}$ can be realized as a planar straight-line embedding in $\mathbb{R}^2$ with edge length $\ell(e)$ for all $e \in E$ is $\exists\mathbb{R}$-complete [Abel et al., JoCG'25], even if $\ell \equiv 1$, but a considerable part of $\mathcal{L}$ is rigid. In this paper, we study the computational complexity of the realization question for structurally simpler, less rigid linkages inside an open polygonal domain $P$, where the placement of some vertices may be specified in the input. We show XP-membership and W[1]-hardness with respect to the size of $G$, even if $\ell \equiv 1$ and no vertex positions are prescribed. Furthermore, we consider the case where $G$ is a path with prescribed start and end position and $\ell \equiv 1$. Despite the absence of any rigid components, we obtain NP-hardness in general, and provide a linear-time algorithm for arbitrary $\ell$ if $G$ has only three edges and $P$ is convex.

Problem

• linkage realization
• polygonal domain
• computational complexity
• planar embedding
• NP-hardness

Innovation

• linkage realization
• polygonal domain
• computational complexity
• NP-hardness
• fixed-parameter tractability