The Parameterized Complexity of Computing the Linear Vertex Arboricity

TL;DR

This work analyzes the parameterized complexity of computing the linear vertex arboricity (LVA) of graphs, which equals the 3D weak line cover number. It proves para-NP-hardness with respect to maximum degree, establishing tight boundaries: graphs with maximum degree at most $4$ (excluding $K_4$) have $LVA\le 2$, while it becomes NP-hard to decide $LVA=2$ for maximum degree $5$; planar graphs have NP-hardness for maximum degree $6$, leaving the degree-$5$ planar case open. It also shows that for any fixed $k\ge1$, deciding whether $LVA\le k$ is fixed-parameter tractable parameterized by treewidth, and provides compact ILP and SAT formulations for practical encoding. The results connect LVA to the broader study of line-cover numbers and establish clear parameterized boundaries, while outlining open questions and directions for future work.

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.

Figures (9)

• Figure 1: The dodecahedron graph has 20 vertices and 30 edges (a), segment number 13, 2D strong line cover number 10 (b), 2D and 3D weak line cover number 2 frw-wlcn-EuroCG18 (c), arc number 10, and circle cover number 5 krw-dgfcf-JGAA19 (d).
• Figure 2: Basic building block $B$
• Figure 3: Variable gadget $V_{a}$ for variable $a$ in $\varphi$
• Figure 4: The gadgets for a two-variable clause $a \lor b$ (left) and for a three-variable clause $a \lor b \lor c$ (right). Both use copies of the basic building block $B$.
• Figure 5: The graph $H_\varphi$ with black clause vertices and white variable vertices (see inset) and the resulting graph $G$ for $\varphi = (\bar{a} \lor \bar{b}) \land (a \lor b \lor c) \land (a \lor c \lor d) \land (\bar{c} \lor d) \land (b \lor \bar{d})$

Theorems & Definitions (19)

• lemma thmcounterlemma
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• lemma thmcounterlemma
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• lemma thmcounterlemma
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• theorem thmcountertheorem
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• lemma thmcounterlemma
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