Constrained Level Planarity is FPT with Respect to the Vertex Cover Number

TL;DR

The paper resolves the parameterized complexity of Constrained Level Planarity (CLP) with respect to vertex cover size $k$, proving an FPT algorithm running in $2^{O(k^2\log k)}\cdot n^{O(1)}$ time. It introduces core-induced subdrawings and refined visibility extensions to bound the essential structure by $O(k)$ vertices/edges, enabling exhaustive but efficient enumeration. The algorithm proceeds in three steps: (1) guess a core, (2) insert transition vertices near visibility edges, and (3) place leaves/ears via a trapezoidal-cell decomposition and traversal sequences, with per-step subroutines guaranteeing extendability to a full CLP drawing. The results establish a tractable frontier for CLP that is tight with respect to several smaller graph parameters and clarifies the limitations for speeding up to polynomial time or extending to related parameters. This advances the understanding of the parameterized landscape of level-planarity variants and suggests directions for studying Partial Level Planarity under VC-based parameters.

Abstract

The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity, each level y is equipped with a partial order <_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of <_y. Constrained Level Planarity is known to be a remarkably difficult problem: previous results by Klemz and Rote [ACM Trans. Alg. 2019] and by Brückner and Rutter [SODA 2017] imply that it remains NP-hard even when restricted to graphs whose tree-depth and feedback vertex set number are bounded by a constant and even when the instances are additionally required to be either proper, meaning that each edge spans two consecutive levels, or ordered, meaning that all given partial orders are total orders. In particular, these results rule out the existence of FPT-time (even XP-time) algorithms with respect to these and related graph parameters (unless P=NP). However, the parameterized complexity of Constrained Level Planarity with respect to the vertex cover number of the input graph remained open. In this paper, we show that Constrained Level Planarity can be solved in FPT-time when parameterized by the vertex cover number. In view of the previous intractability statements, our result is best-possible in several regards: a speed-up to polynomial time or a generalization to the aforementioned smaller graph parameters is not possible, even if restricting to proper or ordered instances.

Figures (2)

• Figure 1: In this (and all other) figure(s), filled square vertices belong to a vertex cover $C$ of the depicted graph and filled (round or square) vertices belong to the core of the shown drawing with respect to $C$. (a) A drawing $\Gamma$ (non-dashed edges) is augmented with visibility edges (dashed) to obtain a visbility extension $\Lambda$ with respect to $C$ (note that this augmentation is not unique). The thick (non-dashed or dashed) edges and filled vertices represent $\Lambda_\mathrm{core}$. (b) All filled round vertices are top crucial ears of $c_a$ and $c_b$. All of them are bounding ears except for $b$ and $c$. The vertex $a$ / $b$ / $c$ / $d$ is an outermost left / outermost right / innermost left / innermost right ear.
• Figure 6: Like in all figures, filled square vertices belong to a vertex cover $C$ of the depicted graph and filled (round or square) vertices belong to the core of the shown drawing with respect to $C$. (a) The drawing $\Lambda^*$; its visibility edges (all between $c_a, c_b$) are dashed. Our goal is to insert the transition vertices of $c_a, c_b$ into the subdrawing $\Lambda_{\mathrm{core}}$ of $\Lambda^*$. The fixed components obtained by deletion of $c_a, c_b$ are $M_1$, consisting of $v_1, c_2, v_7, v_{14}$ and fixed to regions $r_1$ and $r_4$, $M_2$, consisting of $v_3, c_8, v_{16}, v_{17}$ and fixed to region $r_2$, and $M_3$, consisting of $v_4, v_{11}, v_{13}, c_{19}$ and fixed to region $r_3$. The other (non-fixed) components are $v_5, v_6, v_9, v_{10}, v_{12}, v_{15}$, and $v_{18}$; note that these are all either transition vertices or leaves of $c_a, c_b$. (b) The first three boxes show the constraints of $\mathcal{G}$ between the vertices on levels 2, 3, and 4 (only vertices with constraints are shown). The remaining levels have no constraints. The fourth box shows the set $\mathcal{M}$ together with the relation $R$. Note that $M_1$ is not contained in $\mathcal{M}$ since it is fixed to $r_1$ and $r_4$. Further, $R$ contains a cycle. It is removed by the contraction step in which $\mathcal{M}'$ is obtained from $\mathcal{M}$, as depicted in the fifth box. (c) Consider the total order $v_9 \le_{\mathcal{M}'} M_2 \le_{\mathcal{M}'} v_6 \le_{\mathcal{M}'} v_{15} \le_{\mathcal{M}'} v_{18} \le_{\mathcal{M}'} v_{10} \le_{\mathcal{M}'} (M_3 + v_{12}) \le_{\mathcal{M}'} v_5$ on $\mathcal{M}'$. By removing all leaves from $\le_{\mathcal{M}'}$, we obtain the order $M_2 \le_{\mathcal{M}"} v_{15} \le_{\mathcal{M}"} v_{18} \le_{\mathcal{M}"} v_{10} \le_{\mathcal{M}"} (M_3 + v_{12}) \le_{\mathcal{M}"} v_5$ on $\mathcal{M}"$. The figure shows the drawing $\Lambda^{\mathrm t}_{\mathrm{core}}$ obtained by inserting the transition vertices into $\Lambda_{\mathrm{core}}$ according to $\le_{\mathcal{M}"}$. Note that they are not inserted in the same places as in $\Lambda^*$. Further, we cannot simply insert the leaves alongside the transition vertices according to $\le_{\mathcal{M}'}$ since they can have constraints to components in the outer regions $r_1$ and $r_4$ (as, in the example, $v_6$ to $M_1$).

Theorems & Definitions (24)

• Theorem 1
• Lemma 2: DBLP:books/sp/CyganFKLMPPS15
• Corollary 3: Folklore
• Lemma 3: lem:isolated
• Lemma 3: cl:2k-ears-per-level
• Lemma 3: lem:VC-connected
• Lemma 3: lem:size-of-core
• Lemma 3: lem:step1
• Lemma 3: lem:insertTransition
• Lemma 4