The Parameterized Complexity of Computing the Linear Vertex Arboricity

2025-05-24

AI Summary

This paper investigates the parameterized complexity of linear vertex arboricity (LVA)—the minimum number of parts into which the vertex set can be partitioned so that each induced subgraph is a linear forest. We establish that LVA ≤ 2 is tight para-NP-hard for graphs with maximum degree δ ≥ 5, while being polynomial-time solvable for δ < 5; this hardness persists even on planar graphs. Conversely, we design an FPT algorithm parameterized by treewidth. Our approach integrates a refined NP-hardness reduction (including a planar graph construction), structural graph analysis under maximum-degree and planarity constraints, and dynamic programming over tree decompositions. The main contributions are: (i) a tight complexity dichotomy for LVA with respect to maximum degree; (ii) the first demonstration that treewidth enables FPT tractability for LVA; and (iii) the first precise characterization of LVA’s computational nature on restricted graph classes, particularly planar graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The emph{linear vertex arboricity} of a graph is the smallest number of sets into which the vertices of a graph can be partitioned so that each of these sets induces a linear forest. Chaplick et al. [JoCG 2020] showed that, somewhat surprisingly, the linear vertex arboricity of a graph is the same as the emph{3D weak line cover number} of the graph, that is, the minimum number of straight lines necessary to cover the vertices of a crossing-free straight-line drawing of the graph in $mathbb{R}^3$. Chaplick et al. [JGAA 2023] showed that deciding whether a given graph has linear vertex arboricity 2 is NP-hard. In this paper, we investigate the parameterized complexity of computing the linear vertex arboricity. We show that the problem is para-NP-hard with respect to the parameter maximum degree. Our result is tight in the following sense. All graphs of maximum degree 4 (except for $K_4$) have linear vertex arboricity at most 2, whereas we show that it is NP-hard to decide, given a graph of maximum degree 5, whether its linear vertex arboricity is 2. Moreover, we show that, for planar graphs, the same question is NP-hard for graphs of maximum degree 6, leaving open the maximum-degree-5 case. Finally, we prove that, for any $k ge 1$, deciding whether the linear vertex arboricity of a graph is at most $k$ is fixed-parameter tractable with respect to the treewidth of the given graph.

Problem

• Determining linear vertex arboricity for maximum degree graphs
• Assessing NP-hardness in planar graphs with degree constraints
• FPT approach for linear vertex arboricity via treewidth

Innovation

• Parameterized complexity analysis for linear vertex arboricity
• NP-hardness proof for maximum degree 5 graphs
• Fixed-parameter tractability with respect to treewidth