Pathways to Tractability for Geometric Thickness

TL;DR

A full characterization of the problem's parameterized complexity in the extension setting depending on whether the authors parameterize by the number of missing vertices, edges, or both is established.

Abstract

We study the classical problem of computing geometric thickness, i.e., finding a straight-line drawing of an input graph and a partition of its edges into as few parts as possible so that each part is crossing-free. Since the problem is NP-hard, we investigate its tractability through the lens of parameterized complexity. As our first set of contributions, we provide two fixed-parameter algorithms which utilize well-studied parameters of the input graph, notably the vertex cover and feedback edge numbers. Since parameterizing by the thickness itself does not yield tractability and the use of other structural parameters remains open due to general challenges identified in previous works, as our second set of contributions, we propose a different pathway to tractability for the problem: extension of partial solutions. In particular, we establish a full characterization of the problem's parameterized complexity in the extension setting depending on whether we parameterize by the number of missing vertices, edges, or both.

Figures (14)

• Figure 1: Illustration of \ref{['definition:H_and_T']}.
• Figure 2: Illustrations for \ref{['obs:drawing_maps_clones_in_set']} (left), \ref{['lemma:global_crossing_free_region']} (middle, the gray area is the complement of $B$), and \ref{['definition:cells_and_admissible_region_and_clones']} (right, the set $S$ is shown in blue, the cells of $\Gamma$ induced by $S$ are shown in gray).
• Figure 3: A graph $G'$ without degree one and zero vertices (left), a corresponding feedback edge set $F$ and forest $T$ as well as the vertex set $C$ (middle), and the graph $G_0$ (right).
• Figure 4: The drawing $(\Gamma'_i, \chi'_i)$ with $P_{i+1}$ represented as a single red edge (left), and $(\Gamma_{i+1}, \chi_{i+1})$ with resolved crossings and $P_{i+1}$ fully drawn (right). In this example, one red crossing needed to be resolved and $P_{i+1}$ contains 3 internal vertices, hence one additional subdivision was undertaken.
• Figure 5: An instance $(X, k=3)$ of Multicolored Clique (left), the resulting instance of GTE where crosses denote possible vertex positions and non-edges of $X$ are drawn as dashed lines (middle), and a valid extension showing that $X$ contains a $K_3$ (right).

Theorems & Definitions (35)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• definition 1
• definition 2
• lemma 1
• proof
• lemma 2