Point Set Embeddability with List Constraints

2026-07-27

AI Summary

This study addresses the list-constrained planar straight-line embedding problem: given a graph and a point set where each vertex is restricted to a prescribed subset of points, determine whether a planar straight-line embedding exists. By combining computational complexity theory with fixed-parameter tractability (FPT) analysis for both convex and general position point sets, the work establishes a complete complexity landscape. Key contributions include resolving an open question from GD'13 by proving NP-hardness for matchings in convex position, establishing the first NP-hardness result for path embeddings on general point sets, and providing a tight FPT characterization parameterized by vertex cover number. Additionally, the paper presents efficient or parameterized algorithms for several restricted graph classes and demonstrates NP-hardness, APX-hardness, and parameterized intractability for multiple problem variants.

Abstract

Deciding whether a given graph admits a planar straight-line drawing where each vertex is placed on some point from a given finite point set is known as Point Set Embeddability and is a classical problem in graph drawing. In this paper, we study the more general embeddability question where the placement of each vertex $v$ is restricted to a list $L(v)$ of admissible points. We first study the case where the given point set is in convex position. We show that this case is NP-hard even if the given graph is a matching and bi-labeled, i.e., each vertex has at most 2 admissible points. On the positive side, we present two efficient algorithms for the case where the given graph $G$ is connected (and not necessarily bi-labeled): if $G$ is equipped with a combinatorial embedding that needs to be respected, we can solve the problem in polynomial time; otherwise we can solve it in FPT-time with regard to the maximum vertex degree. In particular, this answers an open question by Frati, Glisse, Lenhart, Liotta, Mchedlidze, and Nishat [GD'13]. We then turn our attention to the more general case where the given point set is not necessarily in convex position. Here, we show NP-hardness for bi-labeled paths; notably these graphs have a unique combinatorial embedding and maximum degree two. We also present an FPT-algorithm with respect to the vertex cover number for the special case of bi-labeled graphs. We complement this latter result by establishing paraNP-hardness in the tri-labeled setting for vertex cover number 2 and polynomial-time solvability for vertex cover number 1 and arbitrary $L$. Finally, we study optimization and extension variants, where we want to maximize the number of edges or extend a partial drawing, respectively. For the former, we show APX-hardness and for the latter, we provide a parameterized complexity dichotomy under natural extension parameters.

Problem

• Point Set Embeddability
• List Constraints
• Planar Straight-line Drawing
• NP-hardness
• Parameterized Complexity

Innovation

• Point Set Embeddability
• List Constraints
• Fixed-Parameter Tractability
• NP-hardness
• Graph Drawing