The Parameterized Complexity of Computing the Linear Vertex Arboricity

TL;DR

This paper explores the computational complexity of determining the linear vertex arboricity of graphs, revealing NP-hardness results for various maximum degree classes and establishing fixed-parameter tractability with respect to treewidth.

Contribution

It provides the first parameterized complexity analysis of linear vertex arboricity, showing para-NP-hardness for maximum degree and fixed-parameter tractability for treewidth.

Findings

NP-hard for graphs with maximum degree 5

NP-hard for planar graphs with maximum degree 6

FPT with respect to treewidth for any fixed k

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.