Weakly Leveled Planarity with Bounded Span

TL;DR

This work studies s-span weakly leveled planar drawings, where vertices lie on horizontal levels and edges are either horizontal or strictly $y$-monotone, touching at most $s+1$ levels. It proves NP-hardness for every fixed $s$ and develops FPT algorithms and kernels under structural parameters, including vertex cover and treedepth, as well as a modulatory approach. The paper also provides tight combinatorial bounds for cycle-trees and related classes, showing a constant span bound for 3-connected cycle-trees (span at most $4$) and a logarithmic bound for general cycle-trees, along with SPQ-based tools via path-trees to analyze path- and cycle-structures. These results advance understanding of how structural graph properties constrain layered, monotone drawings and have implications for edge-length ratios and practical drawing algorithms on graphs with bounded vertex cover or treedepth.

Abstract

This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly $y$-monotone curve. A graph is $s$-span weakly leveled planar if it admits such a drawing where the edges have span at most $s$; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing $s$-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter $s$ and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of $2$-outerplanar graphs generalizing Halin graphs, are $Θ(\log n)$-span weakly leveled planar and $4$-span weakly leveled planar when $3$-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration.

Figures (4)

• Figure 1: A $1$-span weakly leveled planar drawing of the Frucht graph (left) and a $4$-span weakly leveled planar drawing of the Goldner-Harary graph (right).
• Figure 6: A $n$-vertex $2$-outerplanar graph requiring $\Omega(n)$ span in every weakly leveled planar drawing.
• Figure 7: (a) An almost-3-connected path-tree $G$, drawn with solid edges. The path-vertices are white and the tree-vertices are black. (b) The path-tree $G$ with root $\rho$.
• Figure 8: Path-trees corresponding to a Q-node (left), an S-node (middle), and a P-node (right). Dashed edges may or may not belong to $G_{\mu}$. Shaded triangles represent smaller path-trees $G_{\nu_i}$.

Theorems & Definitions (23)

• Lemma 2
• Theorem 3
• Lemma 10
• Lemma 11
• Lemma 12
• Theorem 13
• Theorem 14
• Lemma 15
• Lemma 16
• Lemma 17