Clustered Planarity Variants for Level Graphs

TL;DR

The paper investigates clustered planarity in the level-graph setting by introducing two variants, $uCLP$ (unrestricted) and $y$-CLP (y-monotone). It provides a polynomial-time algorithm for $uCLP$ on biconnected single-source graphs by reducing to a Sync CP framework and employing LP-tree/level PQ-tree machinery to enforce level-planar and clustered-planar compatibility, achieving $O(n^3)$ time. Conversely, it proves $y$-CLP is $NP$-hard even under restrictive conditions (single source, biconnected, few levels), via PlanarMonotone3-SAT and 3-Partition reductions with specialized gadgets. The work delineates the landscape of clustered level planarity beyond convex clusters and motivates further study of non-biconnected graphs and potential fixed-parameter tractability results.

Abstract

We consider variants of the clustered planarity problem for level-planar drawings. So far, only convex clusters have been studied in this setting. We introduce two new variants that both insist on a level-planar drawing of the input graph but relax the requirements on the shape of the clusters. In unrestricted Clustered Level Planarity (uCLP) we only require that they are bounded by simple closed curves that enclose exactly the vertices of the cluster and cross each edge of the graph at most once. The problem y-monotone Clustered Level Planarity (y-CLP) requires that additionally it must be possible to augment each cluster with edges that do not cross the cluster boundaries so that it becomes connected while the graph remains level-planar, thereby mimicking a classic characterization of clustered planarity in the level-planar setting. We give a polynomial-time algorithm for uCLP if the input graph is biconnected and has a single source. By contrast, we show that y-CLP is hard under the same restrictions and it remains NP-hard even if the number of levels is bounded by a constant and there is only a single non-trivial cluster.

Figures (7)

• Figure 1: (a) A drawing that is level planar and clustered planar and thus cl-planar, but not convex cl-planar or $y$-cl-planar. (b) A drawing that is $y$-cl-planar (with the augmentation edge in $E'$ shown dashed in red) and thus also cl-planar, but not convex cl-planar.
• Figure 2: (a) A level graph $G$ with two level PQ-trees $T_w$ and $T_v$ derived from (b) its LP-tree. P-nodes are represented by black disks, Q-nodes as white double disks. (c) The graph after replacing $w,v$ by $T_w,T_v$; the orange arrow indicates the sfv-constraint due to $\rho$.
• Figure 3: (a) An instance of Pla-narMono-tone3-SAT. (b) The modified incidence graph. (c) The structure of the corresponding $y$-CLP instance. Highlighted are a clause gadget (top right) and the gadget for propagating variable assignments (bottom right).
• Figure 4: Neither flip of $X'$ admits a valid embedding of the clause gadget if all literals are false.
• Figure 5: The seven variable assignments for which the clause gadget admits a valid embedding.

Theorems & Definitions (20)

• Lemma 0
• Lemma 0
• Lemma 0
• theorem 1
• theorem 2
• proof : Proof Sketch
• theorem 3
• proof : Proof Sketch
• Lemma 3
• proof