[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Henry Förster

University of Tübingen, Tübingen, Germany

As an alternative to previously existing planar graph product structure theorems, we prove that every planar graph G is a subgraph of the strong product of K₂, a path and a planar subgraph of a 4-tree. As an application, we show that the queue number of planar graphs is at most 38 whereas the queue number of planar bipartite graphs is at most 25.

**2012 ACM Subject Classification** Mathematics of computing → Graphs and surfaces; Mathematics of computing → Graph algorithms

**Keywords and phrases** graph product structure, queue layouts, planar graphs

**Acknowledgements** I would like to thank Lena Schlipf and Michael A. Bekos for proofreading an earlier draft version. Further, I want to thank Julia Katheder an Michael Kaufmann for discussions on the topic.

Linear layouts have been a popular research topic in topological graph theory since Bernhart and Kainen [8] introduced book embeddings, which are also known as stack layouts. In a stack layout of a graph, its vertices are placed on a line inducing a linear order ≺ of the vertices, whereas edges are partitioned into sets referred to as pages or stack. As an additional constraint, edges (u, v), (u′, v′) assigned to the same page are forbidden to cross, i.e., we are forbidden to have u ≺ u′ ≺ v ≺ v′. Stack layouts can be used to model chip design processes [11] and thus it is no surprise, that within the decade following their introduction, a tight upper bound for the number of stacks required for embedding planar graphs was found [32]. It is worth remarking that the corresponding lower bound has been established only a few years ago [6, 33] indicating the difficulty of the problem.

As a dual concept to stack layouts, Heath, Leighton and Rosenberg [21, 22] introduced queue layouts. Here, vertices are again placed on a line inducing a linear order ≺, however,

edges are partitioned into queues in which nesting edges (u, v), (u', v') are forbidden, i.e., we do not allow u ≺ u' ≺ v' ≺ v. For a graph G, the minimum number of queues required by any of its queue layouts is called the queue number qn(G). Despite often being highly non-planar, queue layouts still are useful in practical applications such as VLSI [26] and 3D graph drawing [13, 17]. Already Heath, Leighton and Rosenberg [21] conjectured that every planar graph has constant queue number. Despite serious attempts, this conjecture proved elusive for almost 3 decades during which only improved superconstant upper bounds were found [3, 12, 15]. On the other, constant upper bounds for important subclasses of planar graphs [2, 4] provided reasons to believe that a constant upper bound may be possible.

Finally, in 2019, a breakthrough result by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16] proved the correctness of Heath, Leighton and Rosenberg’s conjecture. Namely, they proved that the queue number of planar graphs is at most 49. Potentially even more importantly, the result was obtained using a technique that provided deep insights into the structure of planar graphs. More precisely, they discovered that every planar graph can be expressed as a subgraph of the strong product of a K₃, a path of arbitrary length and a planar graph H of treewidth at most 3. Intuitively, this result shows, that given a BFS-layering of a planar graph, it is possible to partition vertices into a set of bags where 1. the vertices of each bag consist of a so-called tripod, that is, a triangle which is root of three upward paths in the BFS-layering, and 2. the graph obtained by contracting each bag into a single vertex is a planar graph of treewidth at most 3; refer also to Fig. 1a for an illustration. This planar graph product structure theorem also has been successfully applied for coloring applications [7, 18] and already inspired a plethora of follow-up works [7, 10, 14, 18, ?, 24, 27, 29, 30].

Our Contribution. We provide new insights into the structure of planar graphs. An important subclass of planar graphs in our analysis will be the planar quasi-4-trees.

or if G can be constructed from one of the base cases by applying a sequence of the following construction rules:

CR1 G is obtained from a planar quasi-4-tree G′ = (V′, E′) by inserting a new vertex v ∉ V′ and edges (v, t₁) and (v, t₂) and possibly (v, t₃) where (t₁, t₂, t₃) is a triangular face of G′.

CR2 G is obtained from a planar quasi-4-tree G′ = (V′, E′) by inserting a new vertex v ∉ V′ and edges (v, q₁) and (v, q₃) and possibly (v, q₂) and/or (v, q₄) where (q₁, q₂, q₃, q₄) is a quadrangular face of G′.

We remark that it is easy to see from the definition, that planar quasi-4-trees have treewidth at most 4. Moreover, planar 3-trees are a proper subclass of planar quasi-4-trees. Also refer to Figure 2 for an illustration.

As our core result, we prove the following; see Figure 1b:

▶ Theorem 1.2. Let G be a planar graph. Then G admits a H-partition with layered width 2 such that H is a subgraph of a planar quasi-4-tree.

Note that we will formally define H-partitions in Section 2. Theorem 1.2 immediately implies the following graph product structure theorem:

▶ Theorem 1.3. Let G be a planar graph. Then G is subgraph of K₂ ⊠ P ⊠ H where P is a path of arbitrary length and H is a planar quasi-4-tree.

Intuitively, Theorem 1.3 provides an alternative to the result of Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16] where each bag is of simpler structure, namely, a bipod, consisting of an edge which is root of only two upward paths in the BFS-layering. This comes at the expense of increasing the treewidth of graph H from 3 to 4. Thus, it depends on the application whether or not our variant of the planar graph structure product theorem proves more useful.

In particular, we show the usefulness of Theorem 1.3 by investigating the queue number of planar graphs. As a central tool, we prove the following result on the queue number of H, which improves upon a result by Wiechert [31] on the queue number of not necessarily planar graphs of treewidth 4 for the class of planar quasi-4-trees:

▶ Theorem 1.4. Let G be a planar quasi-4-tree. Then, qn(G) ≤ 6.

We believe that this result may be of independent interest as possibly every planar graph of treewidth at most 4 is a subgraph of a planar quasi-4-tree. Combining Theorems 1.2 and 1.4, we slightly improve the best known upper bound [5] on the queue number of planar graphs.

▶ Theorem 1.5. Let G be a planar graph. Then qn(G) ≤ 38.

Finally, we generalize our results to bipartite planar graphs and obtain the following:

▶ Theorem 1.6. Let G be a planar bipartite graph. Then G admits a H-partition with bichromatic layered width 2 such that H is a subgraph of a planar quasi-4-tree.

This allows us to also improve the best known upper bound [20] for the queue number of bipartite planar graphs:

▶ Theorem 1.7. Let G be a planar bipartite graph. Then qn(G) ≤ 25.

Our improvements require adjustments to basically all parts of the proof strategy of Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16] that are on the other hand not too technically difficult, i.e., they "cost just a leg" as the title of our paper suggests. We believe that another nice aspect of our paper - in particular for readers new to the research field - is that consequently it holistically describes the entire proof process whereas [16] was partially using known results [1, 2, 19] as a black box.

## Organization of the Paper.

First, we review necessary definitions and techniques as well as the state-of-the-art in Section 2. Then, in Section 3, we provide our refined approach for layered H-partitions culminating in Theorems 1.2 and 1.3. In Section 4, we investigate the queue number of planar quasi-4-trees yielding Theorem 1.4 which then allows us to prove Theorem 1.5. Finally, we extend our results to bipartite planar graphs in Section 5 obtaining Theorems 1.6 and 1.7. We conclude the paper with open problems in Section 6.

**Planar Graphs and their Drawings.** We require some basic understanding of planar graph drawings. A drawing of a graph \( G = (V, E) \) is a function \( \Gamma : G \rightarrow \mathbb{R}^2 \) that maps each vertex \( v \in V \) to a distinct point \( \Gamma(v) \) of the Euclidean plane and each edge \( (u,v) \in E \) to an open Jordan arc \( \Gamma(u,v) \) connecting \( \Gamma(u) \) and \( \Gamma(v) \). We call a drawing \( \Gamma \) planar if and only if for each pair of edges \( e_1, e_2 \in E, \Gamma(e_1) \) and \( \Gamma(e_2) \) are interior-point disjoint. We call a graph planar if it admits a planar drawing.

Given a planar drawing, the open Jordan arcs representing \( E \) partition the Euclidean plane into a set of regions which we call faces. Planar drawings can be partitioned into equivalence classes called embeddings such that each drawing of the same embedding induces the same set of faces defined by the counter-clockwise cyclic sequence of edges along its boundary. We say a graph is plane if it is associated with a prespecified planar embedding. We also say that we embed a graph, if we compute one of its embeddings.

One face of a planar drawing necessarily is unbounded, we call this face the outer face and the remaining faces internal faces. We call a planar drawing outerplanar if all vertices occur on the outer face. Similarly, we call a graph outerplanar if it admits an outerplanar drawing and outerplane if it is associated with an outerplanar embedding, i.e., a planar embedding where all vertices are incident to the outer face.

Let \( G = (V, E) \) be a plane graph. Given a set \( S \subset V \), the subgraph of \( G = (V, E) \) induced by \( S \) is defined as \( G_S = (S, E \cap (S \times S)) \). An induced subgraph \( G_S \) naturally inherits an induced embedding of \( G \) by considering only the part of the embedding of \( G \) describing \( G_S \).

The dual graph \( D = (V_D, E_D) \) of graph \( G \) contains a vertex for each face in the associated planar embedding of \( G \) and an edge \( (d_1,d_2) \in E_D \) between two vertices \( d_1, d_2 \in V_D \) if the two faces \( f_1 \) and \( f_2 \) of \( G \) corresponding to \( d_1 \) and \( d_2 \) share an edge on their boundaries. The weak dual graph \( D^* \) is obtained from \( D \) by removing the vertex corresponding to the outer face. In particular, the weak dual graph of an outerplane graph is a tree.

Let \( G \) be a planar quasi-4-tree with a known construction sequence using the base cases and construction rules from Definition 1.1. A canonical embedding of \( G \) is constructed as follows. According to the base cases of Definition 1.1 there is an initial \( C_3 \) or \( C_4 \) (Base Cases BC1 and BC2 of Definition 1.1) which serves as the outer face of \( G \). In all remaining steps, we introduce a new vertex \( v \) and connect it to the vertices of a face \( f \) of a graph \( G' \) constructed in a previous step (see Construction Rules CR1 and CR2 of Definition 1.1). In such steps, we embed \( v \) in the interior of \( f \).

**Graph Product Structure.** The first goal of our paper is to express a given planar graph as a subgraph of the strong product of graphs exhibiting an easy-to-describe structure. The strong product \( G \boxtimes H \) of graphs \( G = (V_G, E_G) \) and \( H = (V_H, E_H) \) is formally defined as a graph \( P = (V_P, E_P) \) on vertex set \( V_P = V_G \times V_H \). Its edge set \( E_P \) contains an edge between vertices \( (g_1, h_1) \) and \( (g_2, h_2) \) if and only if 1. \( (g_1, g_2) \in E_G \) and \( h_1 = h_2 \), or, 2. \( g_1 = g_2 \) and \( (h_1, h_2) \in E_H \), or, 3. \( (g_1, g_2) \in E_G \) and \( (h_1, h_2) \in E_H \). Notably, graphs of rather complex structure may be described as products of graphs of easier structure as seen in the example of Figure 3a. Most notably in this regard are the following two theorems known also as Planar Graph Product Structure Theorems:

▶ **Theorem 2.1** (Dujmović, Joret, Micek, Morin, Ueckerdt, Wood [16]). *Let \( G \) be a planar graph. Then \( G \) is subgraph of \( K_3 \boxtimes P \boxtimes H \) where \( P \) is a path of arbitrary length and \( H \) is a planar graph of tree-width at most 3.*

As stated above in Theorem 1.3, we provide yet another such theorem. In comparison to Theorem 2.1, one may at first glance wonder if our result is actually an improvement - namely, we decrease the complexity of the first factor graph while increasing the complexity of the third one. We answer this question positively by showing that our variant of the graph product structure theorem provides an improved result regarding queue layouts.

Queue Layouts. Thus, our second goal is to compute queue layouts of planar graphs that use few queues. A k-queue layout of a graph G = (V, E) is a 2-tuple (≺, Q) where ≺ is a total order of V and Q is a partition of E with |Q| = k such that for each edge set Q ∈ Q for each (u, v), (u′, v′) ∈ Q it holds, that (u, v) and (u′, v′) do not nest, i.e., we do not have u ≺ u′ ≺ v′ ≺ v (see also Figure 3b). For a graph G, the queue number qn(G) is the minimum number k such that G has a k-queue layout. A recent break-through result in this field has been obtained by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16] and improved as follows:

As can be seen from the discussion of our results above, we slightly improve upon these results with Theorems 1.5 and 1.7.

Layered H-Partitions. The central technique used in previous studies [16, 5, 20] for proving graph product structure theorems and constant upper bounds on the queue number of planar graphs are layered H-partitions of graphs; see also Fig 3c.

Let G = (VG, EG) and H = (VH, EH). Further let B = {Bx ⊂ VG|x ∈ VH} be a partition of VG into sets called bag. We say that the 2-tuple (H, B) is a H-partition of G if for each edge (u, v) ∈ EG one of the following holds: 1. u, v ∈ Bx for some x ∈ VH, or 2. u ∈ Bx and v ∈ By for some x, y ∈ VH with x ≠ y but (x, y) ∈ EH. Note that in Case 1, we call (u, v) an intra-bag edge whereas we call (u, v) an inter-bag edge in Case 2. We say that an H-partition has width w, if each bag Vx with x ∈ VH contains at most w vertices.

It turns out that this basic definition of width is often too restrictive in order to abuse it for algorithmic applications. Thus, we define a relaxation using a BFS-layering. A partition L = (L₀, L₁, ...) of V_G is called a BFS-layering of G if Lᵢ contains exactly the vertices with graph-theoretic distance i of a specified vertex r ∈ V_G. Observe that L also partitions E in the sense that each edge (u, v) ∈ E with u ∈ Lᵢ and v ∈ Lⱼ is either an intra-layer edge if i = j or an inter-layer edge if |i − j| = 1. Note that no other edges exist as vertices are layered by graph-theoretic distance. Given a BFS-layering L, we say that an H-partition (H, B) has layered-width ℓ w.r.t. L if |Bₓ ∩ Lᵢ| ≤ ℓ for every x ∈ V_H and i ≥ 0. If additionally there exists no intra-layer edges, we say that H has bichromatic layered width ℓ w.r.t. L.

Intuitively, in order to compute good graph product structure theorem and queue layouts, the goal is to find layered H-partitions that partition G into bags with few vertices on each layer of a BFS-layering such that the bags are connected by a graph H showing nice properties. To this end, all of Theorems 2.1, 2.3, and 2.4 rely on the following result; see also Fig. 1a:

▶ Theorem 2.5 (Dujmović, Joret, Micek, Morin, Ueckerdt, Wood [16]). Let G be a planar graph. Then G admits a H-partition with layered width 3 such that H is a planar graph of treewidth at most 3.

Note that in scope of Theorem 2.5, the content of a bag of the H-partition is known as a tripod. Namely, Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16] construct the bags of an H-partition by finding a triangle t incident to three vertical paths of t in the BFS-layering using Sperner’s lemma [1]. A vertical path P of t here is a path of G containing exactly one vertex v ∈ Lᵢ of t such that either 1. for all vertices w ∈ P \ {v} with w ∈ Lⱼ, we have i > j, or 2. for all vertices w ∈ P \ {v} with w ∈ Lⱼ, we have i < j. As the title of our paper suggests, we will instead compute a layered H-partition such that each bag contains a bipod, that is, an edge e incident to two vertical paths of e in the BFS-layering. For this approach to work, we have to carefully adapt all steps in the proof of Theorem 2.5 to yield Theorem 1.2.

A crucial ingredient in the proof of the graph product structure theorem by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16] is Sperner’s lemma about (not necessarily proper) colorings of internally triangulated graphs. For computing our layered H-partition, we will require a similar result, that allows for more than three colors. We find it worth remarking that a similar result for quadrangulations has been presented by Musin [28, Theorem A] which initially inspired this research until we found the following lemma applicable for triangulations.

▶ Lemma 3.1. Let G = (V, E) be an internally triangulated graph and χ : V → {1, 2, ..., k} a k-coloring of the vertices with k ≥ 3 so that the outer face f₀ of G is (P₁, P₂, ..., P_k) where Pᵢ is a path whose vertices are colored with color i. Then, G contains a triangle (t₁, t₂, t₃) ≠ f₀ such that χ(t₁) = 1, χ(t₂) = 2 and χ(t₃) ∉ {1, 2}.

Proof. We closely follow the proof of Sperner’s Lemma from The Book [1]. Let E₁₂ ⊂ E denote the set of edges whose endvertices have been colored with colors 1 and 2, i.e., for (u, v) ∈ E₁₂ either χ(u) = 1, χ(v) = 2 or χ(u) = 2, χ(v) = 1. Now consider the dual graph

First, assume that \(\deg_{12}(v_1^*) = 3\). Then, all vertices incident to triangle \(t\) must be colored with colors 1 and 2. However, then we have necessarily a monocolored edge which contradicts \(\deg_{12}(v_1^*) = 3\). Thus, we must have \(\deg_{12}(v_1^*) = 1\). Thus, w.l.o.g. \(t = (t_1, t_2, t_3)\) is colored with \(\chi(t_1) = 1\) and \(\chi(t_2) = 2\). Now, if \(\chi(t_3) \in \{1,2\}\), it follows that \(\deg_{12}(t_1^*) = 2\); a contradiction. Thus, \(t_3 \notin \{1,2\}\).

In this section, we prove a stronger version of Theorem 1.2 providing us with the required tools to prove Theorems 1.3 and 1.5. To this end, we more precisely define the following notion of bipods.

Intuitively, we arrive at this definition as follows. In our proof of Theorem 1.2, we will identify bipods by finding a suitable edge \(e\) incident to two vertical paths of \(e\). Now, by definition, a bipod contains at most two vertices on each BFS layer, i.e., the vertices of two consecutive layers may induce a \(K_{2,2}\).

It turns out, that we can assume a simplified structure for our bipods. Namely, we will only consider acyclic bipods, that is bipods containing no cycles. The following lemma proves the validity of this assumption:

Note that for vertex \(v_0^*\) dual to the outer face of \(G\) we have \(\deg_{12}(v_0^*) = 1\) by the coloring of the outer face according to the conditions of the lemma. Since the sum \(\sum_{v^* \in V^*} \deg_{12}(v^*)\) must be even, we have that there exists a vertex \(v_1^* \in V^* \setminus \{v_0^*\}\) with \(\deg_{12}(v_1^*)\) being odd. More precisely, since each vertex \(v^* \in V^* \setminus \{v_0^*\}\) is dual to a triangle \(t\), we have \(\deg_{12}(v^*) \leq 3\), i.e., \(\deg_{12}(v_1^*) \in \{1,3\}\).

\(G^* = (V^*, E^*)\) of \(G\) and let \(E^*_{12} \subset E^*\) denote the set of edges of the dual graph that are dual to edges in \(E_{12}\). For \(v^* \in V^*\) let \(\deg_{12}(v^*)\) denote the number of edges of \(E^*_{12}\) adjacent to \(v^*\).

▶ **Lemma 3.3.** Let \(\mathcal{L} = (L_0, L_1, \ldots)\) be a BFS-layering of a graph \(G = (V, E)\) that is a bipod w.r.t. \(\mathcal{L}\) with disjoint vertical paths \(P_1 = (u_{s_u}, u_{s_u+1}, \ldots, u_{t_u})\) and \(P_2 = (v_{s_v}, v_{s_v + 1}, \ldots, v_{t_v})\) as in Definition 3.2. Then, there exists a non-empty subgraph \(G'\) of \(G\) induced by paths \(P_1' = (u_i, u_{i+1}, \ldots, u_{t_u})\) and \(P_2' = (v_j, v_{j+1}, \ldots, v_{t_v})\) with \(i \geq s_u\) and \(j \geq s_v\) that is an acyclic bipod w.r.t. \(\mathcal{L}\).

▶ **Definition 3.2.** Let \(\mathcal{L} = (L_0, L_1, \ldots)\) be a BFS-layering of a graph \(G = (V, E)\). We call \(G\) bipod if

P1 \(|L_i| \leq 2\) for \(i \geq 0\), and

P2 there are two disjoint vertical paths \(P_1 = (u_{s_u}, u_{s_u+1}, \ldots, u_{t_u})\) and \(P_2 = (v_{s_v}, \ldots, v_{t_v})\) in \(G\) such that a. \(u_i, v_i \in L_i\), b. if both \(P_1\) and \(P_2\) are non-empty \((u_{s_u}, v_{s_v}) \in E\) and c. for each \(v \in V\), \(v\) either belongs to \(P_1\) or \(P_2\).

Proof. By Property P2 of Definition 3.2, we can partition V into two disjoint paths P1 and P2. Assume w.l.o.g. that P1 ≠ ∅. If P2 = ∅, we observe that G is already acyclic as then by Property P2 of Definition 3.2 we actually have |Li| ≤ 1 for i ≥ 0.

Next, assume that P2 ≠ ∅. Choose first i and then j maximally, such that (ui, vj) ∈ E, i.e., ui ∈ Li and vj ∈ Lj and it may be that ui = utu or vj = vtv. Note that (ui, vj) exists by Property P2 of Definition 3.2. We now define G' as the subgraph induced by paths P'1 = (ui, ui+1, ..., utu) and P'2 = (vj, vj+1, ..., vtv). As G' is a subgraph of G, it clearly fulfills Property P1 of Definition 3.2. In addition, since (ui, vj) ∈ E and since P'1 and P'2 are vertical subpaths of P1 and P2 respectively, G' also fulfills Property P2 of Definition 3.2 and thus is a bipod. By the maximal choice of i and j, it follows that (ui, vj) is the only edge connecting a vertex u ∈ P'1 and a vertex v ∈ P'2, thus acyclicity is obtained.

We obtain the following stronger version of Theorem 1.2 where we assume that G is already triangulated and embedded. Note that this assumption is not a loss of generality as we can augment our input graph suitably.

Initialization. We start by computing the H-partition for subgraph G0 consisting only of vertices v1, v2, v3. We initialize H as a triangle (x1, x2, x3) with bags Bxi = {vi} for 1 ≤ i ≤ 3. By Base Case BC1 of Definition 1.1, H is a planar quasi-4-tree, i.e. Invariant I3.4.1 holds. Moreover, each bag Bxi for 1 ≤ i ≤ 3 consists of exactly one vertex. Thus it contains less than 2 vertices on each BFS-layer Li and its induced subgraph is acyclic, i.e., Bxi induces an acyclic bipod and Invariant I3.4.2 is established. In its canonical embedding, H contains exactly one face, namely f = (x1, x2, x3). We specify Cf = (P1, P2, P3) with Pi being the path of length 0 consisting only of vertex vi for 1 ≤ i ≤ 3. As vi ∈ Bxi, we conclude that

Invariant I3.4.3 is established. Further, I3.4.4 hold trivially as H contains no quadrangular face in its canonical embedding. Next, all vertices V \ {v₁, v₂, v₃} of G are located inside the triangle C_f = (v₁, v₂, v₃) as it is cycle bounding the outer face. Since the root of T is located on the outer face, we also have that every vertex v ∈ V \ {v₁, v₂, v₃} is part of a subtree of T rooted at one of v₁, v₂, v₃. Thus, we also have Invariant I3.4.5. Finally, Invariant I3.4.6 follows as both (v₁, v₂, v₃) and (x₁, x₂, x₃) form a clique.

Iterative Step. Here, our aim is to identify another bipod B ≠ ∅ in G that can be added to our H-partition such that Invariants I3.4.1–I3.4.5 are maintained when G_{k+1} is obtained from G_k by adding the vertices of B and all induced edges. We begin by selecting a face f of the canonical embedding of H such that a vertex of V \ {\bigcup_{x∈H} B_x} (i.e., a vertex not assigned to any bipod yet) is contained inside C_f. We then identify a bipod B of G whose vertices are located inside C_f such that 1. if f = (x₁, x₂, x₃) is triangular, there is a vertex b₁ of B incident to a vertex in B_{x₁} and a vertex b₂ of B incident to a vertex in B_{x₂}, 2. if f = (x₁, x₂, x₃, x₄) is quadrangular, there is a vertex b₁ of B incident to a vertex in B_{x₁} and a vertex b₃ of B incident to a vertex in B_{x₃}. To accomplish that, aside from a special degenerate case, we will make use of our generalization of Sperner’s lemma in Lemma 3.1 to identify a triangular face (t₁, t₂, t₃) of G such that t_i is part of a subtree of T rooted at a vertex of B_{x_i} on the boundary of f. In the following, we describe the process in more detail depending on whether f is triangular or quadrangular.

f = (x₁, x₂, x₃) is triangular. For an illustration, refer to Figure 5a. We color each vertex inside C_f and the vertices of C_f such that all vertices of B_{x_i} and all vertices inside C_f which are part of a subtree of T rooted at a vertex of B_{x_i} are colored with color i for 1 ≤ i ≤ 3. Note that by Invariant I3.4.5, this colors all vertices. We prove the following claim:

▷ Claim 3.5. There is an edge (u, v) ∈ E such that w.l.o.g. u is located inside C_f and u and v have different colors.

Proof. Assume for a contradiction that there was no such edge (u, v). By assumption, C_f contains vertices of V \ {\bigcup_{x∈H} B_x}. Hence, assume w.l.o.g. that C_f contains vertices

V1 ⊆ V \ {⋃x∈H Bx} with V1 ≠ ∅ that are colored with color 1. Since there is no edge (u, v) such that u and v have different colors within Cf, we must have that each vertex u ∈ V1 is only incident to other vertices of V1 and to vertices of Bx₁.

Since the subgraph of G inside cycle Cf is internally triangulated, it follows that there exists a single edge (u′, v′) separating the vertices of V1 from all vertices of different colors, i.e., in particular from Bx₂ and Bx₃. This is only possible if both u′, v′ ∈ Bx₁. But then, Bx₁ is not acyclic, a contradiction to Invariant I3.4.2.

We now apply Claim 3.5 to obtain an edge (v₁, v₂) such that w.l.o.g. v₁ and v₂ have colors 1 and 2, respectively. Further, w.l.o.g. v₁ ∉ Bx₁. By Invariant I3.4.5, v₁ is part of a subtree of BFS-tree T rooted at a vertex p₁ of Bx₁. Thus, there exists a path P₁ of length at least one between v₁ and p₁ in T. Similarly, we have that v₂ is part of a subtree of BFS-tree T rooted at a vertex p₂ of Bx₂, i.e., there is a path P₂ of length at least zero between x₂ and p₂ in T. We introduce a new vertex xₖ₊₁ to H and define Bxₖ₊₁ as the subgraph induced by the vertices of P₁ and P₂ without p₁ and p₂. Since v₁ ≠ p₁, we have that |Bxₖ₊₁| ≥ 1. We first observe that Bxₖ₊₁ is a bipod. To this end, observe that each of the paths P₁ and P₂ contains at most one vertex of BFS-layer Li, i.e. they are a vertical path and Bxₖ₊₁ is a bipod.
If Bxₖ₊₁ is not acyclic, by Lemma 3.3 we can find a subset of Bxₖ₊₁ inducing a acyclic bipod such that p₁ and p₂ are connected via a path through that bipod. Thus, Invariant I3.4.2 holds. In H, we connect xₖ₊₁ to x₁ and x₂ and if there is an edge e₃ = (vₖ₊₁, v₃), where vₖ₊₁ ∈ Bxₖ₊₁ and v₃ ∈ Bx₃, also to x₃. Thus, by Construction Rule CR1 of Definition 1.1, H is still a planar quasi-4-tree and Invariant I3.4.1 holds. Moreover, we also immediately yield Invariant I3.4.6. Finally, consider the new faces in the canonical embedding of H. One of the new faces is triangle tₖ₊₁ = (x₁, x₂, xₖ₊₁). Notice that P₁ ∪ P₂ cuts the region inside Cf into two separate parts. One is bounded by a cycle consisting of a path of vertices inside Bx₁, a path of vertices inside Bx₂ and Bₖ₊₁. We define this cycle to be Ctₖ₊₁ = (x₁, x₂, xₖ₊₁).

For the other part, we consider two cases. If e₃ does not exist, it is bounded by a cycle C₄ consisting of a path of vertices inside Bx₁, a path of vertices inside Bx₂, a path of vertices inside Bx₃ and Bₖ₊₁. In addition, we have that the canonical embedding of H contains just one additional new face qₖ₊₁ = (x₁, xₖ₊₁, x₂, x₃). Here, we define C₄ = Cqₖ₊₁. If e₃ does exist, it cuts the cycle C₄ into two subcycles. The first such cycle C₁ consists of a path of vertices inside Bx₁, a path of vertices inside Bxₖ₊₁ and a path of vertices inside Bx₃. The second such cycle C₂ consists of a path of vertices inside Bx₂, a path of vertices inside Bxₖ₊₁ and a path of vertices inside Bx₃. Moreover, in this scenario H contains two additional new faces, namely f₁ = (x₁, xₖ₊₁, x₃) and f₂ = (x₂, xₖ₊₁, x₃). Here, we set Cf₁ = C₁. In both cases, we yield Invariants I3.4.3 and I3.4.4.

Finally, consider any vertex v ∈ V \ {⋃x∈H Bx} contained previously in Cf. It is now contained in a new face f′ of our updated graph H. Moreover, by Invariant I3.4.5, it is part of a subtree of T rooted at a vertex p of Bx₁, Bx₂ or Bx₃. Consider the path Pv between v and p. If Pv does not contain vertices of Bxₖ₊₁, we have that p is contained in a bipod Bx such that x is on the boundary of Cf′. Otherwise, v now is part of a subtree rooted at a vertex of Bxₖ₊₁. Thus, Invariant I3.4.5 is maintained.

f = (x₁, x₂, x₃, x₄) is quadrangular. For an illustration, refer to Figure 5b. We color each vertex inside Cf and the vertices of Cf such that all vertices of Bxᵢ and all vertices inside Cf which are part of a subtree of T rooted at a vertex of Bxᵢ are colored with color i for 1 ≤ i ≤ 4. Note that by Invariant I3.4.5, this colors all vertices. Moreover, by Invariant I3.4.4, this coloring fulfills the preconditions of Lemma 3.1. Thus, w.l.o.g. there exists a triangle

Assuming Assumption 3.6, we have \( t_1 \notin B_{x_1} \) or \( t_3 \notin B_{x_3} \).

Then, assume w.l.o.g. that \( t_1 \notin B_{x_1} \). By Invariant I3.4.5, \( t_1 \) is part of a subtree of BFS-tree \( T \) rooted at a vertex \( p_1 \) of \( x_1 \). Thus, there exists a vertical path \( P_1 \) of length at least one between \( t_1 \) and \( p_1 \) in \( T \). Similarly, we have that \( t_3 \) is part of a subtree of BFS-tree \( T \) rooted at a vertex \( p_3 \) of \( x_3 \), i.e., there is a vertical path \( P_3 \) of length at least zero between \( t_3 \) and \( p_3 \) in \( T \). We introduce a new vertex \( x_{k+1} \) to \( H \) and define \( B_{x_{k+1}} \) as the subgraph induced by the vertices of \( P_1 \) and \( P_3 \) without \( p_1 \) and \( p_3 \). Since \( t_1 \neq p_1 \), we have that \( |B_{x_{k+1}}| \geq 1 \). We first observe that \( B_{x_{k+1}} \) is a bipod. To this end, observe that each of paths \( P_1 \) and \( P_3 \) contains at most one vertex of BFS-layer \( L_i \), i.e., \( B_{x_{k+1}} \) induces a bipod. By Lemma 3.3, we can further assume \( B_{x_{k+1}} \) to induce an acyclic bipod as otherwise we can select a subset of \( B_{x_{k+1}} \) that fulfills this property. In particular, such an acyclic bipod still induces a path that can be extended by \( p_1 \) and \( p_3 \) to obtain a path connecting \( p_1 \) and \( p_3 \). We conclude that Invariant I3.4.2 holds. In \( H \), we connect \( x_{k+1} \) to \( x_1 \) and \( x_3 \). Moreover, if there is an edge \( e_2 = (v_{k+1}, v_2) \), where \( v_{k+1} \in B_{x_{k+1}} \) and \( v_2 \in B_{x_3} \), we also connect \( x_{k+1} \) to \( x_2 \). Similarly, if there is an edge \( e_4 = (v_{k+1}, v_2) \), where \( v_{k+1} \in B_{x_{k+1}} \) and \( v_2 \in B_{x_3} \), we also connect \( x_{k+1} \) to \( x_2 \). Thus, by Construction Rule CR2 of Definition 1.1, \( H \) is still a planar quasi-4-tree and Invariant I3.4.1 holds. Moreover, we also immediately yield Invariant I3.4.6. Finally, consider the new faces in the canonical embedding of \( H \). Notice that \( P_1 \cup P_3 \) cuts the region inside \( C_f \) into two separate parts.

One is bounded by a cycle \( C_{123} \) consisting of a path of vertices inside \( B_{x_1} \), a path of vertices inside \( B_{x_2} \), a path of vertices inside \( B_{x_3} \) and \( B_{k+1} \). If \( e_2 \) does not exist, we have that the canonical embedding of \( H \) contains a new face \( q_{k+1} = (x_1, x_2, x_3, x_{k+1}) \) and we define \( C_{q_{k+1}} = C_{123} \). If \( e_2 \) does exist, it cuts the cycle \( C_{123} \) into two subcycles. The first such cycle \( C_{12} \) consists of a path of vertices inside \( B_{x_1} \), a path of vertices inside \( B_{x_2} \) and a path of vertices inside \( B_{x_{k+1}} \). The second such cycle \( C_{23} \) consists of a path of vertices inside \( B_{x_2} \), a path of vertices inside \( B_{x_3} \) and a path of vertices inside \( B_{x_{k+1}} \). Moreover, in this scenario \( H \) contains two new faces, namely \( f_{12} = (x_1, x_2, x_{k+1}) \) and \( f_{23} = (x_2, x_3, x_{k+1}) \). Here, we set \( C_{f_{12}} = C_{12} \) and \( C_{f_{23}} = C_{23} \).

The second part is bounded by a cycle \( C_{143} \) consisting of a path of vertices inside \( B_{x_1} \), a path of vertices inside \( B_{x_4} \), a path of vertices inside \( B_{x_3} \) and \( B_{k+1} \). If \( e_4 \) does not exist, we have that the canonical embedding of \( H \) contains a new face \( q_{k+1} = (x_1, x_4, x_3, x_{k+1}) \) and we define \( C_{q_{k+1}} = C_{143} \). If \( e_4 \) does exist, it cuts the cycle \( C_{143} \) into two subcycles. The first such cycle \( C_{14} \) consists of a path of vertices inside \( B_{x_1} \), a path of vertices inside \( B_{x_4} \) and a path of vertices inside \( B_{x_{k+1}} \). The second such cycle \( C_{43} \) consists of a path of vertices inside \( B_{x_4} \), a path of vertices inside \( B_{x_3} \) and a path of vertices inside \( B_{x_{k+1}} \). Moreover, in this scenario \( H \) contains two new faces, namely \( f_{14} = (x_1, x_4, x_{k+1}) \) and \( f_{43} = (x_4, x_3, x_{k+1}) \). Here, we set \( C_{f_{14}} = C_{14} \) and \( C_{f_{43}} = C_{43} \).

In all cases, we yield Invariants I3.4.3 and I3.4.4. Finally, consider any vertex \( v \in V \setminus \{ \bigcup_{x \in H} B_x \} \) contained previously in \( C_f \). It is now contained in a new face \( f' \) of our updated graph \( H \). Moreover, by Invariant I3.4.5, it is part of a subtree of \( T \) rooted at a vertex \( p \) of \( B_{x_1} \), \( B_{x_2} \), \( B_{x_3} \) or \( B_{x_4} \). Consider the path \( P_v \) between \( v \) and \( p \). If \( P_v \) does not contain vertices of \( B_{x_{k+1}} \), we have that \( p \) is contained in a bipod \( B_x \) such that \( x \) is on the boundary of \( C_{f'} \). Otherwise, \( v \) now is part of a subtree rooted at a vertex of \( B_{x_{k+1}} \). Thus,

Degenerate Step: Assumption 3.6 does not hold. It remains only to consider the scenario where \( f = (x_1, x_2, x_3, x_4) \) such that Assumption 3.6 does not hold true. In this case, let \( E_{13} \) be the set of edges \((v_1, v_3)\) drawn inside \( f \) such that \( v_1 \in B_{x_1} \) and \( v_3 \in B_{x_3} \). For an illustration, refer to Fig. 6

Observe that \( E_{13} \), which is a crossing-free set of edges, partitions the part of the Euclidean plane inside \( C_f \) into disjoint parts \( P_1, P_2, \ldots, P_p \) with \( |E_{13}| = p - 1 \) where 1. part \( P_1 \) is bounded by a subpath of \( B_{x_1} \), a subpath of \( B_{x_3} \) and a subpath of \( B_{x_2} \) and one edge of \( E_{13} \), 2. part \( P_p \) is bounded by a subpath of \( B_{x_1} \), a subpath of \( B_{x_3} \) and a subpath of \( B_{x_4} \) and one edge of \( E_{13} \), and 3. part \( P_i \) with \( 2 \leq i \leq p - 1 \) is bounded by a subpath of \( B_{x_1} \), a subpath of \( B_{x_3} \), an edge of \( E_{13} \) shared with the boundary of \( P_{i-1} \) and an edge of \( E_{13} \) shared with the boundary of \( P_{i+1} \).

Let \( P_i \) be a non-empty part, i.e., there exists at least one vertex of \( V \setminus \{\bigcup_{x \in H} B_x\} \) inside \( P_i \). Let \( e_{13} \in E_{13} \) be an edge on the boundary of \( P_i \). Since \( G \) is triangulated, there exists a triangle \( t = (t_1, t_2, t_3) \) inside \( P_i \) such that \( (t_1, t_3) = e_{13} \). Because \( P_i \) was chosen to not contain any edge of \( E_{13} \) in its interior and because \( P_i \) is not empty, we conclude that \( t_2 \in V \setminus \{\bigcup_{x \in H} B_x\} \) or if \( i = 1 \) or \( i = p \), we may have that \( t_2 \in B_{x_2} \) or \( t_2 \in B_{x_4} \), respectively. Note that if \( t_2 \in B_{x_2} \) or \( t_2 \in B_{x_4} \), edge \( (t_2, v_y) \) partitions the region \( P_i \) into a region \( P_i^1 \) and a region \( P_i^3 \) such that vertices in \( P_i^1 \) can be incident to vertices from \( B_{x_1} \) but not to vertices from \( B_{x_3} \) whereas vertices in \( P_i^3 \) can be incident to vertices from \( B_{x_3} \) but not to vertices from \( B_{x_1} \). We now augment graph \( G \) by adding a vertex \( v_y \) inside \( t \) and connect \( y \) to all of \( t_1, t_2, t_3 \). Clearly, the resulting graph remains a triangulation.

By Invariant I3.4.5, we have \( t_1 \in L_{i_1}, t_2 \in L_{i_2} \) and \( t_3 \in L_{i_3} \) such that \( i_2 \geq i_3 \) and \( i_2 \geq i_1 \). Assume w.l.o.g. that \( i_1 \leq i_3 \). Then, we augment \( T \) by edge \( (t_1, v_y) \). It is easy to see that \( T \) remains a BFS-tree. Moreover, since \( t \) is a triangle and \( t_1 \) the vertex of \( t \) on the lowest layer, we have \( i_2 - 1, i_3 - 1 \leq i_1 \leq i_2, i_3 \). As \( v_y \) is now added to layer \( L_{i_1 + 1} \), we observe that edges \( (t_2, v_y) \) and \( (t_3, v_y) \) respect the BFS-layering.

We now create a new bag \( B_y = \{v_y\} \). As it contains only one vertex, we trivially fulfill Invariant I3.4.2. Further, in \( H \) we connect \( y \) with \( x_1 \) and \( x_3 \) and if \( t_2 \in B_{x_j} \), also with \( x_j \) for \( j \in \{2, 4\} \). This modification ensures Invariant I3.4.1 as it corresponds to Construction

Invariant I3.4.5 is maintained.

Rule CR2 of Definition 1.1. Moreover, we have also Invariant I3.4.6 as $(t_1, v_y) \in E_G$ and $(t_3, v_y) \in E_G$ but also $(x_1, y) \in E_H$ as well as $(x_3, y) \in E_H$ while in case that $t_2 \in B_{x_j}$, for $j \in \{2,4\}$, we also have $(x_j, y) \in E_H$.

After performing our augmentation for all triangles at the boundary non-empty regions starting with region $P_1$, followed by regions $P_i$ in increasing order of $i$. Moreover, if $2 \leq i \leq p - 1$, we first process the edge of $E_{13}$ shared with $P_{i-1}$. In the canonical embedding of $H$, this partitions face $f$ such that there exists a face $f_i$ containing the bounding bipods of $P_i$. In particular, we yield Invariant I3.4.4 as the vertices of non-empty part $P_i$ are now located in a cycle $C_{f_i}$,
1. bounded by a subpath of $B_{x_1}$, a subpath of $B_{x_3}$ and a subpath of $B_{x_2}$ and a newly inserted bipod $B_y$ if $i = 1, 2$.
2. bounded by a subpath of $B_{x_1}$, a subpath of $B_{x_3}$ and a subpath of $B_{x_4}$ and a newly inserted bipod $B_y$ if $i = p$, and
3. bounded by a subpath of $B_{x_1}$, a subpath of $B_{x_3}$ and two newly inserted bipods $B_y, B_{y'}$ if $2 \leq i \leq p - 1$.

Note that if $i = 1$ or $i = p$ and the new bipod $B_y$ in $P_i$ shares an edge with $B_{x_j}$, where $j = 2$ or $j = 4$, respectively, instead of $f_i$ we yield two triangular faces $f_i^1$ and $f_i^3$ such that $f_i^1$ is bounded by a subpath of $B_{x_1}$, a subpath of $B_{x_j}$ and the newly inserted bipod $B_y$ whereas $f_i^3$ is bounded by a subpath of $B_{x_3}$, a subpath of $B_{x_j}$ and the newly inserted bipod $B_y$. Since we create no other triangular faces, Invariant I3.4.3 follows. Moreover, Invariant I3.4.5 holds as we do not change how vertices of $V_G \setminus (\bigcup_{x\in H} B_x)$ are connected to $T$. Thus, after the degenerate step all invariants are again fulfilled.

Finally, we argue that all of the faces $f_i$ for a non-empty part $P_i$ fulfill Assumption 3.6. To this end, first observe that there is no edge of $E_{13}$ inside $f_i$ as otherwise we would have not defined $P_i$ the way that we did. Thus, it remains to show that the other two bipods bounding $C_f$ are not connected. If $i = 1$ or $i = k$, it is trivial to see that the newly inserted bipod $B_y$ is not incident to a vertex from $B_{x_2}$ or $B_{x_4}$, respectively, as the triangle into which $v_y$ was inserted contains no vertex of $B_{x_2}$ or $B_{x_4}$. If $2 \leq i \leq k - 1$, we observe that the two triangles into which $v_y \in B_y$ and $v_{y'} \in B_{y'}$ are inserted must be disjoint as each such triangle contains exactly one edge of $E_{13}$. Thus, we also have in this scenario that $(v_y, v_{y'}) \notin E$.

## Conclusion

In the paragraphs above, we have described how to perform iterative steps. In particular, we concluded that we can always perform a valid iterative step that satisfies all of our Invariants I3.4.1–I3.4.6. If a degenerate step was performed, we concluded that the number of faces $f$ of $H$ that may cause a degenerate step to happen, decreases. On the other hand, in non-degenerate steps we create at most 2 new quadrangular faces in the canonical embedding of $H$ that may cause a degenerate step. Since in the $k$-th non-degenerate step, at least one vertex of $v \in V \setminus \{\bigcup_{x\in H} B_x\}$ is assigned to a new bipod $B_k$, we conclude that our algorithm assigns all vertices in a linear number of iterations and the theorem follows.

## 3.3 An Alternative Planar Graph Product Structure Theorem

Having established Theorem 1.2, it remains to apply the following lemma, stated as an observation by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16].

▶ Lemma 3.7 (Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16]). Let $G = (V_G, E_G)$ be a graph such that there exists an $H$-partition $(H, B)$ of $G$ with layered width $\ell$ with respect to a BFS-layering $L = (L_0, L_1, \ldots)$ and $H = (V_H, E_H)$. Then, $G$ is a subgraph of $K_\ell \boxtimes P \boxtimes H$ for a path $P$.

For the convenience of the reader we provide a short proof.

**Proof of Lemma 3.7.** Let $d$ be chosen maximally so that $L_d \neq \emptyset$ and let $P = (p_1, p_2, \ldots, p_d)$ and $K_\ell = (V_\ell, V_\ell \times V_\ell)$ with $V_\ell = \{v_1, \ldots, v_\ell\}$. We first show that the graph induced by

each bag is a subgraph of (Kₗ ⊠ P) × ({x}, ∅) for x ∈ V_H. For a fixed layer i ∈ {0, . . . , d}, Kₗ ⊠ P contains an ℓ-clique on ((v₁, p_i), x), ((v₂, p_i), x), . . . , ((v_ℓ, p_i), x). Moreover, between layers i ∈ {0, . . . , d − 1} and i + 1, there is a complete bipartite graph between parts {((v₁, p_i), x), ((v₂, p_i), x), . . . , ((v_ℓ, p_i), x)} and {((v₁, p_{i+1}), x), . . . , ((v_ℓ, p_{i+1}), x)}. Therefore (Kₗ ⊠ P) × ({x}, ∅) includes all possible connections in the bag x.

Next, consider inter-bag edges. According to the definition of H-partitions, there are only connections between two bags B_x and B_y, if (x, y) ∈ E_H. On the other hand, for (x, y) ∈ E_H, we get from the definition of the strong product (Kₗ ⊠ P) ⊠ H, that it contains all edges between ((v_i, p_j), x) and ((v_{i′}, p_{j′}), y) for 1 ≤ i, i′ ≤ ℓ and |j − j′| ≤ 1. Thus, all inter-bag edges are included in the product.

Theorem 1.2 and Lemma 3.7 now immediately prove Theorem 1.3.

This section is devoted to proving Theorem 1.4. We remark that Wiechert [31] proved that not necessarily planar graphs of treewidth 4 have queue number at most 15. In order to do improve this result, we adapt the peeling-into-levels approach used by Alam et al. [2] for computing 5-queue layouts for planar 3-trees to work for planar quasi-4-trees. As a central ingredient, we have to refine a lemma by Dujmović et al. [19, Lemma 22] that discusses layered drawings of triangulated outerplanar graphs for our purposes. To this end, recall that in a layered drawing, each vertex v is placed on a layer ℓ(v) ∈ ℕ such that vertices on the same layer occur on a horizontal line. In a layered drawing, we say that an edge (u, v) has span k if |ℓ(u) − ℓ(v)| = k. Note that while we assume integer y-coordinates, x-coordinates can be real.

For n_f = 1, we select to draw a face of G that corresponds to a leaf in the weak dual of G. Thus, it is either a C₃ or a C₄ and contains at least one edge e_o of the outer face of G. A corresponding drawing where e_o forms the lower envelope is easy to construct; see Figures 7a and 7d, respectively. Hence, assume that n_f > 2. We select a face f of G sharing an edge e_f

Proof. We prove the statement constructively by iteratively drawing a subgraph that contains n_f internal faces of G. We assume w.l.o.g. that G is biconnected as otherwise we can augment G to fulfill our assumption. In addition to the requirements stated in the lemma, we additionally maintain the following invariants:

with the already drawn subgraph \( G_f \) of \( G \) to be drawn next. Note that \( f \) will correspond to a leaf in the weak dual of the subgraph of \( G \) induced by the first \( n_f \) faces, i.e., it shares exactly the single edge \( e_f \) with \( G_f \). Note that we already have obtained a drawing for \( G_f \) meeting the requirements of the lemma and Invariants I4.1.1 and I4.1.2.

Observe that Invariant I4.1.1 allows us to partition the edges on the outer cycle of \( G_f \) into an upper and lower envelope containing the edges visible from \( (0, +\infty) \) and \( (0, -\infty) \), respectively. By Invariant I4.1.2, the lower envelope consists of a single edge that belongs to the outer face, hence \( e_f = (u, v) \) is on the upper envelope. Four cases may arise.

First, if \( \ell(v) - \ell(u) = 1 \) and \( f \) is triangular, define \( t_2 := v, t_1 := u \) and the remaining vertex of \( f \) not belonging to \( G_f \) as \( t_3 \). Then, \( t_3 \) is placed such that \( \ell(t_3) = \ell(t_2) + 1 \) and \( x(t_3) \) is between \( x(t_1) \) and \( x(t_2) \); see Figure 7b.

Second, if \( \ell(v) - \ell(u) = 2 \) and \( f \) is triangular, define \( t_3 := v, t_1 := u \) and the remaining vertex of \( f \) not belonging to \( G_f \) as \( t_2 \). Then, \( t_2 \) is placed such that \( \ell(t_2) = \ell(t_1) + 1 \) and \( x(t_2) \) is between \( x(t_1) \) and \( x(t_3) \); see Figure 7c.

Third, if \( \ell(v) - \ell(u) = 1 \) and \( f \) is quadrangular, define \( q_2 := v, q_1 := u \) and the remaining vertices of \( f \) not belonging to \( G_f \) as \( q_3 \) and \( q_4 \), where \( q_3 \) is neighbor of \( q_1 \) and \( q_4 \) is neighbor of \( q_2 \). Then, \( q_3 \) is placed such that \( \ell(q_3) = \ell(q_2) + 1 \) and \( x(q_3) \) is between \( x(q_1) \) and \( x(q_2) \). Moreover, \( q_4 \) is placed such that \( \ell(q_4) = \ell(q_3) + 1 \) and \( x(q_4) \) is between \( x(q_2) \) and \( x(q_3) \); see Figure 7e.

Finally, if \( \ell(v) - \ell(u) = 2 \) and \( f \) is quadrangular, define \( q_3 := v, q_1 := u \) and the remaining vertices of \( f \) not belonging to \( G_f \) as \( q_2 \) and \( q_4 \), where \( q_2 \) is neighbor of \( q_1 \) and \( q_4 \) is neighbor of \( q_3 \). Then, \( q_2 \) is placed such that \( \ell(q_2) = \ell(q_1) + 1 \) and \( x(q_2) \) is between \( x(q_1) \) and \( x(q_3) \). Moreover, \( q_4 \) is placed such that \( \ell(q_4) = \ell(q_3) + 1 \) and \( x(q_4) \) is between \( x(q_2) \) and \( x(q_3) \); see Figure 7f.

Observe that in all four cases, the new upper envelope is \( x \)-monotone, thus Invariant I4.1.1 holds again. Moreover, the lower envelope remains unchanged, thus Invariant I4.1.2 holds once more. The proof follows. ◀

With Lemma 4.1 established, we describe the peeling-into-levels approach more carefully. Let \( G \) be a planar quasi-4-tree embedded with a canonical embedding. Based on our embedding, we partition \( G \) as follows:

1. Vertices incident to the outer face belong to level set \( L_0 \), also called level zero.

2. Vertices incident to the outer face of the graph induced by \( V \setminus (\bigcup_{k=0}^{i-1} L_k) \) (that is, after deleting level sets \(L_0\) to \(L_{i-1}\)) belong to level set \(L_i\), also called level \(i\).

3. Edges between two vertices of the same level set are called level edges.

4. Edges between two vertices of different levels are called binding edges. Note that the levels of endpoints of such an edge differ by 1.

We will denote by \(G_i\) the subgraph of \(G\) induced by \(L_i\). We now prove that all \(G_i\) have a very simple structure:

▶ Lemma 4.2. Let \(G\) be a canonically embedded planar quasi-4-tree. Then, each connected component of the graph \(G_i\) induced by \(L_i\) is outerplane and each of its internal faces is a triangle or a quadrangle. Further, if \(i > 0\), it is embedded inside a face of \(G_{i-1}\).

Proof. We prove the statement inductively by additionally maintaining the following invariant:

I4.2.1 Each interior face of \(G_i\) is created by one of the construction rules of Definition 1.1 or part of a base case of Definition 1.1.

For graph \(G_0\) the statement of the lemma clearly holds as \(G_0\) is either a \(C_3\) or a \(C_4\). Moreover, also Invariant I4.2.1 is fulfilled as the outer face is defined by Base Case BC1 or BC2 of Definition 1.1.

Thus, assume now that \(G_{i-1}\) has the desired properties and consider a connected component \(C\) of graph \(G_i\). Clearly, \(C\) must be located inside a face \(f\) of \(G_{i-1}\) as the outer face of \(G_{i-1}\) is connected to \(G_{i-2}\) and the embedding of \(G\) is planar. Now, by Invariant I4.2.1, \(f\) is a face constructed by a construction rule of Definition 1.1 or part of a base case of Definition 1.1. Thus, in the fixed construction of \(G\), we insert vertices belonging to \(L_i\) inside of \(f\) by Construction Rules CR1 and CR2 of Definition 1.1. During these operations, we only obtain new faces of \(G\) that are either triangular or quadrangular. We now prove the following fact inductively:

▷ Claim 4.3. Each face \(f'\) created by inserting a vertex of \(L_i\) into \(f\) contains only vertices of \(L_i \cup L_{i-1}\).

Proof. We induce over the number \(n_i\) of vertices of \(L_i\) inserted into \(f\). If \(n_i = 1\), \(f\) is split into two to four faces, each of which containing the one new vertex of \(L_i\) and at least one vertex of \(L_{i-1}\).

Now, assume \(n_i > 1\) and let \(v\) be the \(n_i\)-th vertex of \(L_i\) inserted into \(f\). By induction hypothesis, after inserting the first \(n_i - 1\) vertices of \(L_i\) into \(f\), we only created faces \(F'\) containing only vertices of \(L_i\) and \(L_{i-1}\). As \(v\) belongs to \(L_i\) and thus must be visible from a vertex of \(L_{i-1}\), it must be inserted into a face \(f' \in F'\) that contains a vertex of \(L_{i-1}\). As \(v\) is inserted according to Construction Rules CR1 and CR2 of Definition 1.1, this procedure only creates faces that only contain vertices of \(L_i \cup L_{i-1}\). ◁

The proof follows now immediately from Claim 4.3 as each face created by a Construction Rule of Definition 1.1 is a triangle or quadrangle. ◀

We now have the prerequisite tools for establishing Theorem 1.4.

Proof of Theorem 1.4. We refine the proof of Alam et al. [2] for computing 5-queue layouts for planar 3-trees. A schematic illustration of our approach can be found in Figure 8. Let \(L_0, \ldots, L_\lambda\) denote the levels of graph \(G\). We incrementally compute a queue layout \((\prec, Q)\) of the graph induced by \(L_0 \cup \ldots \cup L_i\) with \(i \in \{0, \ldots, \lambda\}\). During this process, we maintain the following invariants:

We begin by showing how to construct a 2-queue layout \((\prec_C, \{L_1, L_2\})\) of each connected component \(C\) of \(G_i\); see dark red and black edges in Fig. 8b. By Lemma 4.2, we know that \(C\) is an outerplanar graph with each internal face being a triangle or a quadrangle. Thus, by Lemma 4.1, we find a layered drawing \(\Gamma_C\) of \(C\). Let \(\prec_i\) denote the horizontal left-to-right order of vertices on layer \(i\) in \(\Gamma_C\). We obtain \(\prec_C\) by concatenating \(\prec_1, \prec_2, \dots, \prec_h\) where \(h\) denotes the height of \(\Gamma_C\). Further, we assign each edge of \(C\) with span 1 to \(L_1\) and each edge of \(C\) with span 2 to \(L_2\). By Property P1 of \(\Gamma_C\) according to Lemma 4.1, we have assigned all edges of \(C\).

Proof. Assume for a contradiction that two edges \((u, v), (u', v')\) assigned to \(L_i\) nest. Assume w.l.o.g. that \(\ell(u) = \ell(v) - i\) and \(\ell(u') = \ell(v') - i\). First, assume that \(\ell(u) \neq \ell(u')\), say w.l.o.g. that \(\ell(u) < \ell(u')\). Then we have by construction that \(u \prec_C u'\) and since the span of \((u, v)\) and \((u', v')\) are both \(i\) also \(\ell(v) < \ell(v')\). It follows by construction that \(v \prec_C v'\) and \((u, v)\) and \((u', v')\) cannot nest.

Thus, it must be that \(\ell(u) = \ell(u')\). Assume w.l.o.g. that \(u \prec_C u'\), in particular, we even have that \(u \prec_{\ell(u)} u'\). In order for \((u, v)\) and \((u', v')\) to nest, we have \(v' \prec_C v\). Since \(\ell(v) = \ell(v')\), we also have \(v' \prec_{\ell(v)} v\). But this contradicts the planarity of the layered drawing \(\Gamma_C\) as \((u, v)\) and \((u', v')\) are straight-line, start and end at the same layers, respectively, but have to swap horizontal order in-between these layers.

For \(L_0\), we note that there is only a single connected component \(C\) and we use the obtained 2-queue layout. Thus, we establish Invariant I1.4.3 whereas Invariants I1.4.1, I1.4.2 and I1.4.4 are trivially fulfilled. Moreover, the theorem holds for \(\lambda = 0\).

I1.4.2 There is a linear order \(\prec_{c(G_i)}\) of the connected components of \(G_i\) such that for each pair \(C_1\) and \(C_2\) of such components with \(C_1 \prec_{c(G_i)} C_2\) we have for each pair of vertices \(c_1 \in C_1\) and \(c_2 \in C_2\) that \(c_1 \prec c_2\).

For the inductive step, we first have to reconsider Properties P2 and P3 of Γ_C according to Lemma 4.1. Namely, consider an internal face f of Γ_C. All of its incident vertices are located on different layers according to Lemma 4.1. We define the layer-i-vertex v_i(f) with i ∈ {1, ..., 4} as follows.

\[
v_i(f) = \begin{cases}
t_i & \text{if } f \text{ is a triangle and } i \leq 3 \\
\emptyset & \text{if } f \text{ is a triangle and } i = 4 \\
q_i & \text{if } f \text{ is a quadrangle}
\end{cases}
\]

The following claim significantly simplifies the following discussion.

▷ Claim 4.5. There exists a total order ≺_{f(C)} of the faces of C such that if f ≺_{f(C)} f' for two faces f and f' of C, we have v_i(f) ≺_C v_i(f') for i ∈ {1, ..., 4} and v_i(f), v_i(f') ≠ ∅.

Proof. We construct ≺_{f(C)} as follows. We set f ≺_{f(C)} f' if v_1(f) ≺_C v_1(f') or if v_1(f) = v_1(f') and f occurs to the left of f' in the left-to-right order of faces incident to v_1(f) in Γ_C. It remains to discuss that also v_i(f) ≺_C v_i(f') if f ≺_{f(C)} f' for i ∈ {2, ..., 4}.

Note that by definition, we have ℓ(v_1(f)) ≤ ℓ(v_1(f')). If ℓ(v_1(f)) < ℓ(v_1(f')), we obtain also ℓ(v_i(f)) < ℓ(v_i(f')) and thus v_i(f) ≺_C v_i(f'). Thus, it remains to consider the case where ℓ(v_1(f)) = ℓ(v_1(f')). In this scenario consider edges (v_1(f), v_i(f)) and (v_1(f'), v_i(f')). Note that if i = 4 and f and f' are quadrangular, these edges can be inserted as y-monotone curves into face f and f', respectively, and do not occur outside of these faces as C is outerplanar (refer also to the blue dashed edges in Fig. 7). Edges (v_1(f), v_i(f)) and (v_1(f'), v_i(f')) occur in between layers ℓ(v_1(f)) and ℓ(v_i(f)) as crossing-free y-monotone curves in Γ_C. Thus, if v_1(f) ≠ v_1(f'), it follows from v_1(f) ≺_{ℓ(v_1(f))} v_1(f') that also v_i(f) ≺_{ℓ(v_1(f))+(i-1)} v_i(f'). Otherwise, v_1(f) = v_1(f') and (v_1(f), v_i(f)) and (v_1(f'), v_i(f')) occur in the same left-to-right order around v_1(f) as f and f'. Thus, the claim follows. ◁

The following claim significantly simplifies the following discussion.

We are now ready to describe the incremental step of our algorithm. We process the connected components of graph G_i in the left-to-right order ≺_C(G_i). For the connected component C, we process its faces in the left-to-right order ≺_{f(C)}. For face f of C, we check if it contains connected components of G_{i+1}. If so, we compute a 2-queue layout of each of these components as described above and concatenate its linear order to ≺. Then, we assign all binding edges incident to v_i(f) to queue B_k for k ∈ {1, ..., 4}; see red, blue, green and orange edges in Fig. 8b, respectively. We then proceed with the next face or component in ≺_{f(C)} or ≺_C(G_i), respectively. Clearly, this incremental step maintains Invariants I1.4.1, I1.4.2 and I1.4.3 for each new component inserted. Thus, it remains to discuss that the binding edges between L_i and L_{i+1} maintain Invariant I1.4.3. We first consider the case, where different components of G_i are involved.

▷ Claim 4.6. Let (u, v) and (u', v') be two binding edges between L_i and L_{i+1} such that u belongs to a component C of G_i and u' belongs to a component C' of G_i. Then, (u, v) and (u', v') do not nest.

Proof. Assume w.l.o.g. that C ≺_C(G_i) C'. By Invariant I1.4.3, we also have u ≺ u'. By Lemma 4.2 each connected component C' of G_{i+1} is connected via binding edges to vertices of exactly one component C of G_i. As the components of G_{i+1} are inserted according to the left-to-right order ≺_C(G_i) it also follows that v ≺ v' and the statement follows. ◁

Now consider the case where both edges are part of the same connected component.

▷ Claim 4.7. Let (u, v) and (u′, v′) be two binding edges between Li and Li+1 such that u and u′ belong to a component C of Gi. If (u, v), (u′, v′) ∈ Bk for k ∈ {1, . . . , 4}, (u, v) and (u′, v′) do not nest.

Proof. In this scenario, by Lemma 4.2 we know that each component C′ of Gi+1 is inserted into a face f of Gi. Since (u, v), (u′, v′) ∈ Bk, we have u = vk(f) and u′ = vk(f′) for faces f and f′ of Gi. Moreover, v is part of a component inserted into f whereas v′ is part of a component inserted into f′. If f = f′, it follows that u = u′ and no nesting occurs. Otherwise, assume w.l.o.g. that u ≺ u′. Then, also f ≺f(C) f′ by Claim 4.5 and therefore v ≺ v′ since we inserted the components of Gi+1 that have binding edges incident to vertices of C according to order ≺ (f(C)). ◁

Claims 4.6 and 4.7 imply Invariant I1.4.4 and the theorem follows as Q = {L1, L2, B1, B2, B3, B4} has cardinality six. ◀

## 4.2 General Planar Graphs

Using Theorem 1.2 and Theorem 1.4, we could immediately apply the following lemma by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16] to obtain an upper bound of 39:

▶ Lemma 4.8 (Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16]). Let G = (VG, EG) be a graph such that there exists an H-partition (H, B) of G with layered width ℓ with respect to a BFS-layering L = (L0, L1, . . .) and H = (VH, EH). Then,

\[ qn(G) \leq 3\ell \cdot qn(H) + \left\lfloor\frac{3\ell}{2}\right\rfloor. \]

However, we can slightly improve the argumentation used to proof Lemma 4.8 to show Theorem 1.5. In anticipation of Theorem 1.7, we prove the following slightly stronger variant of Theorem 1.5.

▶ Theorem 4.9. Let G be a planar graph. Then qn(G) ≤ 38. In particular, if G is a triangulated embedded plane graph with a predetermined BFS-layering L rooted at a vertex of the outer face, it admits a 38-queue layout (≺, Q) such that there is a subset Q* ⊂ Q of cardinality |Q*| = 13 containing only intra-layer edges w.r.t. L.

Proof. We closely follow the proof of Lemma 4.8 by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [16]. First, we apply Theorem 3.4 to identify a H-partition (H, B) of layered width 2 where each B ∈ B induces an acyclic bipod. We first discuss how to compute a 1-queue layout (≺x, {Qb}) of the induced subgraph of each Bx ∈ B with x ∈ H as follows. By Definition 3.2, we can partition the vertices of Bx into two disjoint vertical paths P1x = (ux_sx, ux_sx+1, . . . , ux_tx) and P2x = (vx_σx, vx_σx+1, . . . , vx_τx) such that sx ≤ σx. We now define ≺x such that uxi ≺x vxi for sx ≤ i ≤ max{tx, τx} and αi ≺x βj with sx ≤ i ≤ j ≤ max{tx, τx} and α, β ∈ {ux, vx}. We assign all edges induced by Bx to queue Qb. See Figure 9a for an illustration.

▷ Claim 4.10. (≺x, Qb) is a 1-queue layout of the subgraph of G induced by Bx.

Proof. Assume for a contradiction that two edges of (u, v) and (u′, v′) induced by Bx nest. Assume w.l.o.g. that u ≺x u′, u ≺x v and u′ ≺x v′. We first observe that (u, v) and (u′, v′) cannot belong to the same vertical path Pix with i ∈ {1, 2} as the vertices of Pix occur in the order induced by Pix in ≺x. Thus, we necessarily obtain u ≺x v ≺x u′ ≺x v′, i.e. (u, v) and (u′, v′) do not nest.

- DRAFT -

We now construct the queue layout (≺, Ω) of H. To this end, we first compute a 6-queue layout (≺_H, ℋ) of H with ℋ = {H₁, H₂, ..., H₆}. We now define ≺ as follows. If u ∈ Lᵢ and v ∈ Lⱼ with j ≥ i, then u ≺ v. Otherwise, u, v ∈ Lᵢ. Then, if u ∈ Bₓ and v ∈ Bᵧ with x ≠ y, we set u ≺ v if and only if x ≺_H y. Otherwise, we also have u, v ∈ Bₓ and set u ≺ v if and only if u ≺ₓ v. We assign the edges of G to queues of Ω as follows:

Second, assume that (u, v) = (uₛₓˣ, v_σₓˣ). Then, we always have u ≺ₓ v ≺ₓ u′ ≺ₓ v′ unless sₓ = σₓ − 1 and (u′, v′) = (u_σₓˣ, v_σₓ₊₁ˣ). In this scenario we obtain u ≺ₓ u′ v ≺ₓ ≺ₓ v′. Thus in both cases, (u, v) and (u′, v′) do not nest.

Thus, we necessarily have (u, v) ∈ Pᵢˣ and (u′, v′) ∈ Pⱼˣ with i ≠ j and i, j ∈ {1, 2}. If i = 2, u ≺ u′ implies that the edges are (u, v) = (vₐˣ, v_{a+1}ˣ) and (u′, v′) = (v_bˣ, v_{b+1}ˣ) with s_{uₓ} ≤ a < b ≤ max{t_{uₓ}, t_{vₓ}}. Thus, we have either u ≺ₓ v ≺ₓ u′ ≺ₓ v′ if a < b − 1 or u ≺ₓ u′ ≺ₓ v ≺ₓ v′, i.e. no nesting. Otherwise, we i = 1 and u ≺ u′ implies that the edges are (u, v) = (uₐˣ, u_{a+1}ˣ) and (u′, v′) = (v_bˣ, v_{b+1}ˣ) with s_{uₓ} ≤ a ≤ b ≤ max{t_{uₓ}, t_{vₓ}}. Thus, we have either u ≺ₓ v ≺ₓ u′ ≺ₓ v′ if a < b or u ≺ₓ u′ ≺ₓ v ≺ₓ v′, i.e. no nesting. This contradicts our assumption that (u, v) and (u′, v′) nest.

We claim that each queue contains no nestings:

Proof. Assume for a contradiction that there are two edges (u, v), (u′, v′) ∈ Q_b^i that nest. By Claim 4.10, we know that two (u, v) and (u′, v′) belong to different bags, i.e., u, v ∈ Bₓ and u′, v′ ∈ B_y. Assume w.l.o.g. that u ≺ u′, u ≺ v and u′ ≺ v′. If u, u′ ∈ Lᵢ, we obtain from

≺_H that also v ≺ v′ and (u, v) and (u′, v′) do not nest. Hence, we have u ∈ L_i, u′ ∈ L_j and i < j. But then, we also have that v ∈ L_{i′} and v′ ∈ L_{j′} with i′ < j′ and hence v ≺ v′. Thus (u, v) and (u′, v′) do not nest, a contradiction.

▷ Claim 4.12. The edges of X^α_h do not nest for 1 ≤ h ≤ 6, X ∈ {L, F, B} and α ∈ {s, ℓ}.

Proof. Assume for a contradiction that there are edges (u, v), (u′, v′) ∈ X^w_h that nest. W.l.o.g. assume that u ≺ u′, u ≺ v and u′ ≺ v′. First, observe that if u ∈ L_i and u′ ∈ L_j with i < j, we have that v ∈ L_i and v′ ∈ L_j, i.e., u ≺ v ≺ u′ ≺ v′, i.e. (u, v) and (u′, v′) do not nest. Thus, we have u, u′, v, v′ ∈ L_i. Let u ∈ B_x, u′ ∈ B_{x′}, v ∈ B_y and v′ ∈ B_{y′}. Clearly, we have x ≺_H y and x′ ≺_H y′. If y = x′, we have that v ≺ v′ and no nesting occurs. Thus, y ≠ x′. Moreover, by u ≺ u′, we also cannot have y′ = x.
If x ≠ x′ and y ≠ y′, we obtain that the order of u, u′, v, v′ in ≺ corresponds to the order of x, x′, y, y′ in ≺_H. As edges (x, y), (y, y′) do not nest in H_h, it follows that neither do (u, v) and (u′, v′).
If x = x′ and y = y′, we observe that only edges (u^x_i, v^y_{i+k}) and (v^x_i, u^y_{i+k}) nest where k = 0 if X = L, k = 1 if X = F and k = −1 if B = L. But one of these edges belongs to X^ℓ_h while the other one belongs to X^s_h.
If x = x′ but y ≠ y′, we necessarily have u = u′ = u^x_{i+k} where k = 0 if X = L, k = 1 if X = F and k = −1 if B = L. Thus we must have x ≠ x′ but y = y′. However, then v = v′ = v^y_{i+k}; a contradiction to the fact that (u, v) and (u′, v′) must be non-adjacent.

Since we have shown that no queue contains a nesting, we realize that the obtained linear layout is a 38-queue layout with queues Q^ℓ_b, Q^i_b and for 1 ≤ h ≤ 6 L^s_h, L^ℓ_h, F^s_h, F^ℓ_h, R^s_h and R^ℓ_h. The proof follows by observing that L ∪ Q^ℓ_b contain only intra-layer edges where |L ∪ Q^ℓ_b| = 13.

▶ Lemma 5.1 (Förster, Kaufmann, Merker, Pupyrev and Raftopoulou [20]). Let G = (V, E) be a quadrangulated plane graph and r a vertex on the outer face of G. Then, there exists a triangulated supergraph G* = (V, E ∪ E_T) of G and a BFS-tree T rooted at r inducing a BFS-layering L = (L_0, L_1, ...) of G* such that each edge of E_T is a intra-layer edge w.r.t. L.

For the sake of self-containment of this paper, we explicitly state here the following proof provided in [20]:

Proof of Lemma 5.1 [20]. To compute T we perform a BFS-traversal of G starting from vertex r. Observe that in the BFS-traversal, for each quadrangle q = (q_1, q_2, q_3, q_4) we have that w.l.o.g. q_1 ∈ L_i, q_2, q_4 ∈ L_{i+1}. We add edge (q_2, q_4) to triangulate q, i.e. q_2, q_4 ∈ E_T. Since (q_2, q_4) is an intra-layer edge, the lemma follows.

Finally, we are ready to prove Theorem 1.6 and Theorem 1.7.

We conclude the paper by stating intriguing open problems motivated by our results.

Proof of Theorems 1.6 and 1.7. We start by computing a quadrangulated supergraph of \( G \). Then, we apply Lemma 5.1 and obtain a triangulated supergraph \( G^* \) of \( G \) as well as a BFS-tree \( T \) rooted at the outer face. We can then apply Theorems 3.4 and 4.9 and obtain our results since by Lemma 5.1 all intra-layer edges are only part of \( G^* \) but not of \( G \).

OP2 In conjunction with Open Problem OP1, we also remark that planar graphs of treewidth 4 received little attention in the literature so far in general. On the other hand, there exist several results on planar 3-trees in the literature; see e.g. [2, 9, 23, 25]. Theorem 1.3 also motivates to study planar graphs of treewidth 4 more closely. A nice concrete problem in this direction would be to settle the following conjecture:

We remark that the \( (4 \times k) \)-grid graph is not a counterexample for Conjecture 6.1.

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