An Improved Upper Bound on the Queue Number of Planar Graphs

Abstract

A k-queue layout is a special type of a linear layout, in which the linear order avoids $(k+1)$-rainbows, that is, $k+1$ independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, which is defined as the minimum value of k for which a k-queue layout is feasible. Recently, Dujmović et al. [J. ACM, 67(4), 22:1–38, 2020] showed that the queue number of planar graphs is at most 49, thus settling in the positive a long-standing conjecture by Heath, Leighton and Rosenberg. To achieve this breakthrough result, their approach involves three different techniques: (1) an algorithm to obtain 2-queue layouts of outerplanar graphs, (2) an algorithm to obtain 5-queue layouts of planar 3-trees, and (3) a decomposition of a planar graph into so-called tripods. In this work, we push further each of these techniques to obtain the first non-trivial improvement of the upper bound on the queue number of planar graphs from 49 to $42$.

1 Introduction

Linear layouts of graphs have a long tradition of study in different contexts, including graph theory and graph drawing, as they form a framework for defining different graph-theoretic parameters with several applications; see, e.g., [1]. In this regard, one seeks to find a total order of the vertices of a graph that reaches a certain optimization goal [2,3,4]. In this work, we focus on a well-studied type of linear layouts, called queue layout [5,6,7,8], in which the goal is to minimize the size of the largest rainbow, namely, a set of independent edges that are pairwise nested. Equivalently, the problem asks for a linear order of the vertices and a partition of the edges into a minimum number of queues (called queue number), such that no two independent edges in the same queue are nested [7]; see Fig. 1b.

Queue layouts of graphs were introduced by Heath and Rosenberg [7] in 1992 as the counterpart of stack layouts (widely known also as book embeddings), in which the edges must be partitioned into a minimum number of stacks (called stack number), such that no two edges in the same stack cross [9]; see Fig. 1c. Since their introduction, queue layouts of graphs have been a fruitful subject of intense research with several important algorithmic and combinatorial results proposed over the years [6, 8, 10,11,12,13,14]; for an introduction, we refer the reader to [15].

The most intriguing problem in this research field is undoubtedly the problem of specifying the queue number of planar graphs, that is, the maximum queue number over all planar graphs. This problem dates back to 1992, when Heath, Leighton and Rosenberg conjectured that the queue number of planar graphs is bounded [6]. Notably, despite the different efforts [11, 16, 17], this conjecture remained unanswered for more than two decades. That only changed in 2019 with a breakthrough result of Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [5, 18], who managed to settle in the positive the conjecture, as they showed that the queue number of planar graphs is at most 49. The best-known corresponding lower bound is 4 due to Alam et al. [10].

It is immediate to see, however, that the gap between the currently best-known lower and upper bounds is rather large, which implies that the exact queue number of planar graphs is, up to the point of writing, still unknown. Note that this is in contrast with the stack number of planar graphs, which was recently shown to be exactly 4 [19, 20]. Also, the existing gap in the bounds on the queue number of planar graphs gives the intuition that it is unlikely the upper bound of 49 by Dujmović at al. [5] is tight, even though in the last two years no improvement appeared in the literature.

Fig. 1

a A planar straight-line drawing of the octahedron graph and b–c different linear layouts of it

1.1 Our Contribution

We verify the aforementioned intuition by reducing the upper bound on the queue number of planar graphs from 49 to $42$ (see Theorem 7 in Sect. 5). This improvement requires non-trivial adjustments to each of the three algorithms that are involved in the approach by Dujmović at al. [5], namely: (i) an algorithm to obtain 2-queue layouts of outerplanar graphs by first computing straight-line drawings of them in which the y-distance of any two adjacent vertices is either 1 or 2 [21]; see Sect. 4.1, (ii) an algorithm to obtain 5-queue layouts of planar 3-trees [10]; see Sect. 4.2, and (iii) a decomposition of a planar graph into so-called tripods [5]; see Sect. 4.3. Although we assume familiarity with these algorithms, in Sect. 3 we outline their most important aspects. For preliminary notions and standard terminology refer to Sect. 2.

2 Preliminaries

A drawing of a graph maps each vertex to a distinct point of the Euclidean plane and each edge to a Jordan curve connecting its endpoints. A drawing of a graph is planar, if no two edges cross (except at common endpoints). A graph that admits a planar drawing is called planar; see Fig. 1a. A planar drawing partitions the plane into topologically connected regions called faces; its unbounded one is called outer face. A combinatorial embedding of a planar graph is an equivalence class of planar drawings that are pairwise topologically equivalent, that is, they define the same set of faces up to an orientation-preserving homeomorphism of the plane. A planar graph together with a combinatorial embedding is a plane graph. A graph that admits a planar drawing, in which all vertices are incident to its outer face, is called outerplanar. It is known that outerplanar graphs have treewidth at most 2. A planar graph with treewidth at most 3 is commonly referred to as a planar 3-tree. Planar 3-trees are subgraphs of maximal planar 3-trees, which are recursively defined as follows. In the base case, the graph is a 3-cycle. Otherwise, there exists a vertex of degree 3 whose removal yields a maximal planar 3-tree with one vertex less.

A linear layout of a graph is defined by a total order of its vertices and an objective over its edges. We denote the (total) vertex order of a simple undirected graph G by $\prec$, such that for any two vertices u and v of G, $u \prec v$ if and only if u precedes v in the order. Let F be a set of $k \ge 2$ independent edges $(u_i, v_i)$ of G, that is, $F=\{(u_i, v_i):\;i=1,\ldots ,k\}$. If $u_1\prec \ldots \prec u_k\prec v_k\prec \ldots \prec v_1$, then we say that the edges of F form a k-rainbow, while if $u_1\prec v_1\prec \ldots \prec u_k\prec v_k$, then the edges of F form a k-necklace. The edges of F form a k-twist, if $u_1\prec \ldots \prec u_k\prec v_1\prec \ldots \prec v_k$; see Fig. 2. Finally, two independent edges that form a 2-rainbow (2-necklace, 2-twist) are referred to as nested (disjoint, crossing, respectively).

Fig. 2

Illustration of a a 3-rainbow, b a 3-necklace, and c a 3-twist

There are several types of linear layouts of graphs in the literature, with queue layouts being among the most studied of them. Formally, a k-queue layout of a graph G consists of a vertex order $\prec$ of G and a partition of the edges of G into k sets of pairwise non-nested edges with respect to $\prec$, called queues. A preliminary result by Heath and Rosenberg [7] states that a graph admits a k-queue layout if and only if it admits a vertex order in which no $(k+1)$-rainbow is formed. The queue number of a graph G, denoted by $\mathrm {qn}(G)$, is the minimum k, such that G admits a k-queue layout. Accordingly, the queue number of a class of graphs is the maximum queue number over all its members.

3 Outline of the Involved Techniques

In this section, we summarize the main aspects of the three algorithms mentioned in Sect. 1 that are involved in the approach by Dujmović et al. [5] to achieve the bound of 49 on the queue number of planar graphs. The first is an algorithm by Dujmović, Pór, and Wood [21] to compute 2-queue layouts of outerplanar graphs (Sect. 3.1). The second one is by Alam et al. [10] to compute 5-queue layouts of planar 3-trees (Sect. 3.2). The last one is the actual algorithm by Dujmović et al. [5] to compute 49-queue layouts of planar graphs (Sect. 3.3).

3.1 Outerplanar Graphs

As already stated, the main ingredient of the algorithm by Dujmović, Pór and Wood [21] is the construction of a straight-line drawing $\Gamma (G)$ of a maximal outerplane graph G that can be transformed into a 2-queue layout of G. The recursive construction of $\Gamma (G)$ maintains the following invariant properties:

• (O.1) The cycle delimiting the outer face consists of two strictly x-monotone paths, referred to as upper and lower envelopes, respectively.

• (O.2) The y-coordinates of the endvertices of each edge differ by either 1 (span-1 edge) or 2 (span-2 edge).

Fig. 3

Introducing a degree-2 vertex v along the upper envelope, when its two neighbors u and w are connected with a a span-1 edge, and b a span-2 edge

To maintain Invariants (O.1) and (O.2), Dujmović et al. adopt an approach according to which, at each recursive step, a vertex of degree 2 is added to the already constructed drawing; see Fig. 3. The base of the recursion consists of a triangle¹ that can be trivially drawn with two span-1 edges and one span-2 edge satisfying Invariants (O.1) and (O.2). Assume that G has $n > 3$ vertices. Since G is biconnected outerplane, it contains a vertex v of degree 2. Removing v yields a biconnected outerplane graph $G'$ with $n-1$ vertices, which admits a drawing $\Gamma (G')$ satisfying Invariants (O.1) and (O.2). By Invariant (O.1), none of the edges on the outer face is drawn vertically in $\Gamma (G')$. To obtain drawing $\Gamma (G)$ of G, vertex v is inserted into $\Gamma (G')$ as follows. Let u and w be the neighbors of v in G. Since G is maximal outerplane, (u, w) is an edge of G that lies on the outer face of $\Gamma (G')$. By Invariant (O.2), (u, w) is either a span-1 or a span-2 edge. Assume that (u, w) is along the upper envelope of $\Gamma (G')$; the case where it is along the lower envelope is symmetric. Assume first that (u, w) has span 1 in $\Gamma (G')$ and without loss of generality that $y(u)=y(w)+1$. Then, vertex v is placed such that $y(v)=y(u)+1$ and $x(v)=\frac{1}{2}(x(u)+x(w))$; see Fig. 3a. Hence, (u, v) and (v, w) have span 1 and 2, respectively, which implies that Invariant (O.2) is maintained. Since (the non-vertical) edge (u, w) of the upper envelope of $\Gamma (G')$ is replaced by the x-monotone path $u \rightarrow v \rightarrow w$ in $\Gamma (G)$, Invariant (O.1) is also maintained. Assume now that (u, w) has span 2 in $\Gamma (G')$ and without loss of generality that $y(u)=y(w)+2$. Then, vertex v is placed such that $y(v)=y(u)-1$ and $x(v)=\frac{3x(w)+x(u)}{4}$; see Fig. 3b. This implies that both edges (u, v) and (v, w) have span 1. Similarly to the previous case, one can argue that Invariants (O.1) and (O.2) are maintained.

Finally, drawing $\Gamma (G)$ is transformed into a 2-queue layout of G as follows:

• (i) for any two vertices u and v of G, $u \prec v$ if and only if either $y(u) > y(v)$, or $y(u)=y(v)$ and $x(u)<x(v)$ in $\Gamma (G)$, and

• (ii) the edges assigned to the first (second) queue in the layout are those with span 1 (span 2, respectively) in $\Gamma (G)$.

3.2 Planar 3-trees

Alam et al. [10] adopt a peeling-into-levels approach [22] to produce a 5-queue layout of a maximal plane 3-tree H. Initially, the vertices of H are partitioned into levels $L_0,\ldots ,L_\lambda$ with $\lambda \ge 1$, such that $L_0$-vertices are incident to the outer face of H, while $L_{i+1}$-vertices are in the outer face of the subgraph of H obtained by the removal of all vertices in $L_0,\ldots ,L_i$, for $i=0,\ldots ,\lambda -1$. The edges of H are partitioned into level and binding edges, depending on whether their endpoints are on the same or on consecutive levels respectively. Let $H_i$ be the subgraph of H induced by the edges of level $L_i$, where $i=0,\ldots ,\lambda$. Since H is a maximal plane 3-tree, each connected component of $H_i$ is an internally triangulated outerplane graph. Therefore, it is embeddable in two queues. This implies that each connected component c of $H_{i+1}$ (which is outerplane) lies in the interior of a triangular face of $H_i$. As a result there are exactly three vertices of $H_i$ that are connected to vertices of c. The construction of the 5-queue layout of H satisfies the following invariant properties:

• (T.1) In the linear order ${\prec _\mathrm {H}}$, all vertices of level $L_i$ precede all vertices of level $L_{i+1}$ for every $i=0,\ldots ,\lambda -1$;

• (T.2) Vertices of each connected component of level $L_i$ appear consecutively in ${\prec _\mathrm {H}}$ for every $i=0,\ldots ,\lambda$;

• (T.3) All level edges of H are assigned to two queues denoted by ${\mathcal {Q}}_0$ and ${\mathcal {Q}}_1$;

• (T.4) For every $i=0,\ldots ,\lambda -1$, the binding edges between vertices of $L_i$ and $L_{i+1}$ are assigned to three queues ${\mathcal {Q}}_2$, ${\mathcal {Q}}_3$ and ${\mathcal {Q}}_4$ as follows. For each connected component c of $H_{i+1}$, let x, y and z be its three neighbors in $H_i$ so that $x\;{\prec _\mathrm {H}}\; y\;{\prec _\mathrm {H}}\; z$. Then, the binding edges between $L_i$ and $L_{i+1}$ with one endpoint in c are assigned to ${\mathcal {Q}}_2$, ${\mathcal {Q}}_3$ or ${\mathcal {Q}}_4$ if their other endpoint is x, y or z, respectively.

3.3 General Planar Graphs

Dujmović et al. [5] present an algorithm to produce a 49-queue layout of a (general) plane graph. Central in their algorithm is the notion of an H-partition², defined as follows. Given a graph G, an H-partition of G is a partition of the vertices of G into sets $A_x$ with $x\in V(H)$, called bags, such that for each edge (u, v) of G with $u\in A_x$ and $v \in A_y$ either $x=y$ holds or (x, y) is an edge of H. In the former case, (u, v) is called intra-bag edge, while in the latter case it is called inter-bag edge. A BFS-layering of G is a partition $\mathcal {L}=(V_0,V_1,\ldots )$ of its vertices according to their distance from a specific vertex of G. In other words, it is a special type of H-partition, where H is a path and each bag $V_i$ corresponds to a layer. In this special type of H-partition, an intra-bag edge is called intra-layer, while an inter-bag edge is called inter-layer³. An H-partition has layered-width $\ell$ with respect to a BFS-layering $\mathcal {L}$ if each bag of H has at most $\ell$ vertices on each layer of $\mathcal {L}$.

The following lemma is the main tool of the algorithm by Dujmović et al. [5], since it yields a construction of a queue-layout of a graph G, whose number of queues depends on the layered-width of an H-partition of G and the queue number of H.

Lemma 1

(Dujmović et al. [5]) For all graphs G and H, if H admits a k-queue layout and G has an H-partition of layered-width $\ell$ with respect to some layering $\mathcal {L}=(V_0,V_1,\ldots )$ of G, then G admits a $\left( 3\ell k+\lfloor \frac{3}{2}\ell \rfloor \right)$-queue layout using vertex order $\overrightarrow{V_0},\overrightarrow{V_1},\dots$, where $\overrightarrow{V_i}$ is some order of $V_i$. In particular,

$$\begin{aligned} \mathrm {qn}(G) \le 3 \ell \,\mathrm {qn}(H) + \lfloor \tfrac{3}{2}\ell \rfloor . \end{aligned}$$

(1)

In the original proof of Lemma 1 (see [5, Lemma 9]), the order of the vertices of G on each layer of $\mathcal {L}$ is defined as follows. Let $x_1, \dots , x_h$ be the vertices of H as they appear in a k-queue layout $QL (H)$ of H, that is, $x_1{\prec _\mathrm {H}}\; \dots {\prec _\mathrm {H}}\; x_h$, and let $A_{x_1}, \dots , A_{x_h}$ be the corresponding bags of the H-partition. Then, the order $\overrightarrow{V_i}$ of each layer $V_i$ of $\mathcal {L}$ with $i\ge 0$ is:

$$\begin{aligned} \overrightarrow{V_i}=A_{x_1}\cap V_i, A_{x_2}\cap V_i, \dots , A_{x_h}\cap V_i \end{aligned}$$

where each subset $A_{x_j}\cap V_i$ is ordered arbitrarily, for $j=1,\ldots , h$. This gives the total order ${\prec _\mathrm {G}}$ for the vertices of G. The edge-to-queue assignment, which completes the construction of the queue layout $QL (G)$ of G, exploits the following two properties:

• (P.1) Two intra-bag edges of G nest in ${\prec _\mathrm {G}}$, only if they belong to the same bag; see blue and red edges in Fig. 4a.

• (P.2) Two inter-layer edges of G nest in ${\prec _\mathrm {G}}$, only if their endpoints belong to the same pair of consecutive layers of $\mathcal {L}$; see blue, purple and orange edges in Fig. 4.

Recall that the edges of G are classified into four categories given by the bags of the H-partition and the layers of $\mathcal {L}$, namely, intra-layer intra-bag edges, inter-layer intra-bag edges, intra-layer inter-bag edges and inter-layer inter-bag edges. The edge-to-queue assignment is based on these four categories. We start with the intra-bag edges of G, that is, edges whose endpoints belong to the same bag; see Fig. 4a.

• (E.1) Intra-layer intra-bag edges of G are assigned to at most $\lfloor \frac{\ell }{2} \rfloor$ queues, as the queue number of $K_\ell$ is $\lfloor \frac{\ell }{2} \rfloor$ [22]; see red edges in Fig. 4a.

• (E.2) Inter-layer intra-bag edges of G are assigned to at most $\ell$ queues, as the queue number of $K_{\ell , \ell }$ is $\ell$, when all vertices of the first bipartition precede those of the second; see blue edges in Fig. 4a.

Fig. 4

Illustration of a Intra-bag edges; the intra-layer ones are red, while the inter-layer ones are blue, and b inter-bag edges; the intra-layer ones are green, while the inter-layer ones are purple (forward) and orange (backward). In both subfigures, a bag is illustrated as a gray box, in which vertices of different layers (that are part of the bag) are drawn at different horizontal lines; the white vertices indicate that the intersection of a layer with the content of a bag may consist of more than four vertices; the edges towards these vertices have not been illustrated (Color figure online)

The remaining edges of G are the inter-bag edges that connect vertices of different bags; see Fig. 4b. Inter-layer inter-bag edges are further partitioned into two categories. Let (u, v) be an inter-layer inter-bag edge with $u \in A_x \cap V_i$ and $v \in A_y \cap V_{i+1}$, for some $i\ge 0$ and $x\ne y$. Then (u, v) is forward, if $x\; {\prec _\mathrm {H}}\; y$ holds in $QL (H)$, otherwise, (u, v) is backward; see purple and orange edges in Fig. 4b, respectively. For all inter-bag edges, in total, $3\ell k$ queues suffice.

• (E.3) Intra-layer inter-bag edges of G are assigned to at most $\ell k$ queues; on each layer, an edge of H corresponds to a subgraph of $K_{\ell , \ell }$, where all vertices of the first bipartition precede those of the second; see green edges in Fig. 4b.

• (E.4) Forward inter-layer inter-bag edges of G are assigned to at most $\ell k$ queues; for two consecutive layers, an edge of H corresponds to a subgraph of $K_{\ell , \ell }$, where all vertices of the first bipartition precede those of the second; see purple edges of Fig. 4b.

• (E.5) Symmetrically all backward inter-layer inter-bag edges of G are assigned to at most $\ell k$ queues; see orange edges in Fig. 4b.

The next property follows from the edge-to-queue assignment of inter-bag edges of G. Details can be found in the proof of Lemma 9 in [5].

• (P.3) For $1 \le i \le r$, let $(u_i,v_i)$ be an edge of G, such that $u_i\; {\prec _\mathrm {G}}\; v_i$, $u_i \in A_{x_i}$ and $v_i \in A_{y_i}$. If all these r edges belong to one of (E.3)-(E.5) and form an r-rainbow in ${\prec _\mathrm {G}}$, while edges $(x_1,y_1),\ldots ,(x_r,y_r)$ of H are assigned to the same queue in $QL (H)$, then $r \le \ell$ and either $u_1,\ldots ,u_r$ or $v_1,\ldots , v_r$ belong to the same bag of the H-partition of G.

In order to bound the parameters k and $\ell$ used in Lemma 1, the fact that G is maximal plane is being exploited. To this end, a few more ingredients are needed. A vertical path of G in a BFS-layering $\mathcal {L}$ is a path $P= v_0,\dots , v_\kappa$ of G consisting only of edges of the BFS-tree of $\mathcal {L}$ and such that if $v_0$ belongs to $V_i$ in $\mathcal {L}$, then $v_j$ belongs to $V_{i+j}$, with $j=1,\ldots ,\kappa$. Further, we say that $v_0$ and $v_\kappa$ are the first and last vertices of P. A tripod of G consists of up to three pairwise vertex-disjoint vertical paths in $\mathcal {L}$ whose last vertices form a clique of size at most 3 in G. We refer to this clique as the base of the tripod. Dujmović et al. [5] describe a recursive algorithm that partitions the vertices of G into tripods, whose bases are triangular faces of G. In particular, they show that for any BFS-layering $\mathcal {L}$, G admits an H-partition with the following properties:

• (P.4) H is a planar 3-tree and thus $QL (H)$ is a k-queue layout with $k \le 5$ [10].

• (P.5) Its layered-width $\ell$ is at most 3, since each bag induces a tripod in G.

Properties (P.4) and (P.5) along with Eq. (1) imply that the queue number of planar graphs is at most $3 \cdot 3 \cdot 5 + \lfloor \tfrac{3}{2}\cdot 3 \rfloor =49$.

4 Refinements of the Involved Techniques

In this section, we present refinements of the algorithms outlined in Sect. 3 that will allow us to reduce the upper bound on the queue number of planar graphs from 49 to $42$.

4.1 Outerplanar Graphs

The initial algorithm by Dujmović et al. [21], outlined in Sect. 3.1, produces outerplanar straight-line drawings that satisfy Invariants (O.1) and (O.2). Our modification ensures two additional properties that are stated in Lemma 2. To achieve this, besides (O.1) and (O.2), we maintain a third invariant:

• (O.3) The lower envelope consists of a single edge.

To maintain Invariant (O.3), we observe that a biconnected maximal outerplane graph G with at least four vertices, contains at least two non-adjacent degree-2 vertices. Let x be such a degree-2 vertex of G, which we assume to be fixed in the recursive construction of $\Gamma (G)$. In particular, by our previous observation, we can always remove a degree-2 vertex that is different from x at every recursive step. This ensures that x will eventually appear in the triangle $\mathcal {T}$ that is drawn at the base of the recursion. We draw $\mathcal {T}$, such that x is its bottommost vertex while its two incident edges are of span-1 and span-2 as follows. Let y and z be the other two vertices of $\mathcal {T}$. We draw x, y and z at points (2, 0), (0, 1) and (1, 2), respectively. This implies that (x, y) forms the lower envelope of $\mathcal {T}$, while (y, z) and (z, x) form the upper one. This guarantees that Invariant (O.3) is maintained at the base of the recursion. Let $G'$ be the subgraph of G obtained by removing a degree-2 vertex v from G with $v\ne x$. We recursively compute a drawing $\Gamma (G')$ of $G'$ that satisfies Invariant (O.3), such that the edge (x, y) of $\mathcal {T}$ forms the lower envelope of $\Gamma (G')$. Since (x, y) belongs to the outer face of G, it follows that v is incident to two vertices of the upper envelope of $\Gamma (G')$. So, after the addition of v to $\Gamma (G')$ in order to obtain $\Gamma (G)$, the lower envelope of $\Gamma (G)$ consists only of edge (x, y), as desired. This concludes the construction of $\Gamma (G)$ satisfying all three Invariants (O.1), (O.2) and (O.3).

In the following, we focus on proving two properties of drawing $\Gamma (G)$, that are summarized in Lemma 2. Let $\langle u,v,w \rangle$ be a face of $\Gamma (G)$ such that $y(u)-y(w)=2$ and $y(u)-y(v)=y(v)-y(w)=1$. We refer to vertices u, v and w as the top, middle and bottom vertex of the face, respectively⁴.

Lemma 2

Let G be a biconnected maximal outerplane graph, and let $\Gamma (G)$ be an outerplanar drawing of G satisfying Invariants (O.1)–(O.3). Then, each vertex of G is

• (a) the top vertex of at most two triangular faces of $\Gamma (G)$, and

• (b) the middle vertex of at most two triangular faces of $\Gamma (G)$.

Proof

For (a), consider a vertex u of G. If u is the top vertex of a face, then u is incident to a span-2 edge (u, v) with $y(u)>y(v)$. By Invariant (O.3), u is a successor of v in the recursive approach by Dujmović et al. [21], namely, when u is placed in the recursively constructed drawing, vertex v belongs to its upper envelope. We now claim that u cannot be incident to two span-2 edges (u, v) and $(u,v')$ with $y(u)>y(v)$ and $y(u)>y(v')$, that is, with the properties mentioned above; this claim implies part (a) of the lemma. Assuming the contrary, by Invariant (O.2), when u is placed in the drawing at most one edge incident to u has span 2. So, at most one of (u, v) and $(u,v')$ is drawn when u is placed in the drawing, which implies that at least one of v or $v'$, say $v'$, is a successor of u. Thus, $y(u)<y(v')$ holds; a contradiction.

For (b), assume for a contradiction that G contains a vertex u, which is the middle vertex of three triangular faces, say $\mathcal {T}_1$, $\mathcal {T}_2$ and $\mathcal {T}_3$, of $\Gamma (G)$. For a significantly small constant $\varepsilon >0$, let $p_1=(x(u)-\varepsilon ,y(u))$ and $p_2=(x(u)+\varepsilon ,y(u))$, and consider the two horizontal rays $r_1$ and $r_2$ emanating from vertex u, such that $r_1$ and $r_2$ contain points $p_1$ and $p_2$, respectively. Since u is middle vertex for $\mathcal {T}_1$, $\mathcal {T}_2$ and $\mathcal {T}_3$, it follows by Invariant (O.2) that the span-2 edge of each of $\mathcal {T}_1$, $\mathcal {T}_2$ and $\mathcal {T}_3$ crosses either $r_1$ or $r_2$; by the choice of $\varepsilon$, we may assume that if a span-2 edge of one of the faces $\mathcal {T}_1$, $\mathcal {T}_2$ and $\mathcal {T}_3$ crosses $r_1$ ($r_2$), then point $p_1$ ($p_2$, respectively) lies in the interior of this face. On the other hand, by outerplanarity, each of the points $p_1$ and $p_2$ can be contained in at most one of $\mathcal {T}_1$, $\mathcal {T}_2$ and $\mathcal {T}_3$; a contradiction. $\square$

Note that part (a) of Lemma 2 requires Invariant (O.3), that is, it does not necessarily hold for all drawings obtained by the algorithm by Dujmović et al. [21]. On the other hand, part (b) of Lemma 2 is a property of the original algorithm as it holds for drawings that do not necessarily satisfy Invariant (O.3).

4.2 Planar 3-trees

To maintain Invariant (T.3), Alam et al. [10] use the algorithm by Dujmović et al. [21] to assign the level edges of $L_0,\ldots ,L_\lambda$ of the input plane 3-tree H to two queues ${\mathcal {Q}}_0$ and ${\mathcal {Q}}_1$, since on each level these edges induce a (not necessarily connected) outerplane graph. In our approach we adopt the modification for the algorithm by Dujmović et al. [21] introduced in Sect. 4.1. As Invariants (O.1) and (O.2) are preserved, queues ${\mathcal {Q}}_0$ and ${\mathcal {Q}}_1$ still suffice for the level edges of H. We exploit Invariant (O.3), that our modification additionally satisfies, to derive an important property of the produced queue layout, as stated in Lemma 3.

To maintain Invariant (T.4), Alam et al. [10] adopt the following assignment scheme for the binding edges between $L_j$ and $L_{j+1}$ to queues ${\mathcal {Q}}_2$, ${\mathcal {Q}}_3$ and ${\mathcal {Q}}_4$, for each $j = 0, \ldots , \lambda -1$. Consider a binding edge (u, v) with $u \in L_j$ and $v \in L_{j+1}$. Then, vertex u belongs to a connected component $C_u$ of the subgraph $H_j$ of H induced by the level-$L_j$ vertices. Similarly, vertex v belongs to a connected component $C_v$ of the corresponding subgraph $H_{j+1}$ of H. Furthermore, $C_u$ is outerplane and its 2-queue layout has been computed by the algorithm by Dujmović et al. [21], while $C_v$ resides in the interior of a triangular face $\mathcal {T}_v$ of $C_u$ in the embedding of H, such that u is on the boundary of $\mathcal {T}_v$. In the edge-to-queue assignment scheme by Alam et al. [10], edge (u, v) is assigned to ${\mathcal {Q}}_2$, ${\mathcal {Q}}_3$ or ${\mathcal {Q}}_4$ if and only if u is the middle, top or bottom vertex of $\mathcal {T}_v$, respectively [10]. For a vertex $u\in L_j$ with $j=0,\ldots , \lambda -1$, we denote by $N_2(u)$, $N_3(u)$ and $N_4(u)$ the neighbors of u in $L_{j+1}$ such that the edges connecting them to u are assigned to queues ${\mathcal {Q}}_2$, ${\mathcal {Q}}_3$ and ${\mathcal {Q}}_4$, respectively. The next property follows from Lemma 2.

Lemma 3

For $j = 0, \ldots , \lambda -1$, let $u\in L_j$ be a vertex of H in our modification of the peeling-into-levels approach by Alam et al. [10]. Then, the vertices of $N_2(u)$ precede those of $N_3(u)$ in ${\prec _\mathrm {H}}$. Also, the vertices of $N_2(u)$ belong to at most two connected components of $H_{j+1}$ (residing within distinct faces of $H_{j}$); the same holds for the vertices of $N_3(u)$.

Proof

The first part of the lemma is proven in [10], although it is not explicitly stated as part of Invariants (T.1)–(T.4). For the second part, consider a binding edge (u, v) with $v\in N_2(u)$; a similar argument applies when $v \in N_3(u)$. By the definition of $N_2(u)$, vertex u is the top vertex of the triangular face $\mathcal {T}_v$ of $H_j$, in which the connected component $C_v$ of $H_{j+1}$ that contains v resides. Since by Lemma 2a vertex u can be the top vertex of at most two triangular faces of $H_j$, there exist at most two connected components of $H_{j+1}$, to which vertex v can belong. $\square$

4.3 General Planar Graphs

Dujmović et al. [5] recursively compute the bags (that is, the tripods) of the H-partition. In this recursive approach, each newly discovered tripod $\tau$ is adjacent to at most three other tripods $\tau _1$, $\tau _2$ and $\tau _3$ already discovered. We say that $\tau _1$, $\tau _2$ and $\tau _3$ are the parents of $\tau$; see Fig. 5a. Also, each non-empty vertical path of $\tau$ is connected to only one of its parents via an edge of the BFS-tree used to construct the BFS-layering $\mathcal {L}$ (black in Fig. 5a). This property gives rise to at most three sub-instances (gray in Fig. 5a), which are processed recursively to compute the final tripod decomposition.

Fig. 5

Tripod $\tau$ with parents $\tau _1$, $\tau _2$ and $\tau _3$. In a $\tau$ shares no vertex with $\tau _1$, $\tau _2$, $\tau _3$ and in b $\tau$ shares a vertex with $\tau _3$

Note that, in general, one or more vertical paths of a tripod may have no vertices; see Fig. 5b. We refer to such a tripod as degenerated. In addition, we call a degenerated tripod empty if it shares a vertex with each of its parents. Hence, empty tripods have no vertices and as a result do not contribute to H. To avoid considering degenerated tripods that are not empty explicitly, we introduce an appropriate augmentation of the input graph. More precisely, let f be the base triangle of a non-empty degenerated tripod $\tau$, and let T be the BFS tree that was used to compute $\tau$. We augment G by adding a triangle $f'$ in the interior of face f. By an appropriate triangulation, we can obtain a triangulated plane graph $G'$ that contains G as a subgraph, such that all vertices of $f'$ are leaves of T, which gives rise to a BFS tree $T'$ of $G'$. In this way, face f of G gets three leaves of $T'$ inside it, one attached to each vertex of f. Hence, the base triangle of the corresponding tripod in $G'$ is not degenerated. This allows us to assume without loss of generality in the following that, in the tripod decomposition, either each tripod is empty or non-degenerated, that is, no vertical path of it is empty.

Fig. 6

Illustration for Properties (P.6) and (P.8). Dotted and dashed edges would introduce crossings

For $i=1,2,3$, let $p_i^1$, $p_i^2$ and $p_i^3$ be the three vertical paths of tripod $\tau _i$. Up to renaming, we assume that $\tau$ lies in the cycle bounded by (parts of) $p_1^1$, $p_1^2$, $p_2^1$, $p_2^2$, $p_3^1$ and $p_3^2$ as in Fig. 5a. The next properties follow by planarity and the BFS-layering:

• (P.6) There is no edge connecting a vertex of $\tau$ to a vertex of $p_i^3$ for $i=1,2,3$; see the dotted edge in Fig. 6.

• (P.7) Let $v_i^p$ be the vertex of vertical path p of $\tau$ on layer $V_i$ of $\mathcal {L}$. For two vertical paths p and q of $\tau$, edge $(v_i^p,v_j^{q})$ belongs to G only if $\vert i-j\vert \le 1$.

• (P.8) For vertical paths p and q of $\tau$, at most one of the edges $(v_i^p,v_{i+1}^q)$ and $(v_{i+1}^p,v_{i}^q)$ exists in G; see the dashed edges of Fig. 6.

Note that Properties (P.6), (P.7), and (P.8) hold even if $\tau$ has less than three parents, or if the cycle bounding the region of $\tau$ does not contain two vertical paths of each parent tripod.

In the original algorithm by Dujmović et al. [5], each vertex $v_\tau$ in H corresponds to a non-empty tripod $\tau$ in G, and an edge $(v_\tau ,v_{\tau '})$ exists in H, if $\tau$ is a parent of $\tau '$ in G, or vice versa. By construction, H is a connected partial planar 3-tree, which is arbitrarily augmented to a maximal planar 3-tree $H'$. Then, the algorithm by Alam et al. [10] is employed to compute a 5-queue layout of $H'$ as required by Lemma 1. Here, we adopt a particular augmentation to guarantee an additional property for the graph $H'$ (see Lemma 4). Similarly to the original approach, we contract the vertices of each non-empty tripod $\tau$ of G to a single vertex $v_\tau$. However, in our modification, we keep self-loops that occur when an edge of G has both endpoints in $\tau$ (unless this edge belongs to one of its vertical paths), as well as, parallel edges that occur when two vertices of $\tau$ have a common neighbor not in $\tau$. Two important properties of this contraction scheme that follow directly from planarity are given below; see Fig. 7.

• (P.9) The edges around each contracted vertex $v_\tau$ appear in the same clockwise cyclic order as they appear in a clockwise traversal along $\tau$ in G.

• (P.10) The edges having at least one endpoint on the same vertical path of $\tau$ appear consecutively around $v_\tau$; see Fig. 7b.

Fig. 7

a A tripod $\tau$ in G where the edges incident to its three vertical paths are drawn dotted, and b the result after contracting $\tau$ to $v_\tau$

Note that the aforementioned contraction scheme creates self-loops and pairs of parallel edges. We focus on homotopic self-loops and pairs of parallel edges, which contain no vertex either in the interior or in the exterior region that they define. We remove such self-loops and keep one copy of such parallel edges. Then, we subdivide each (non-homotopic) self-loop twice, and for each edge with multiplicity $m>1$, we subdivide all but one of its copies. In this way, each vertex $v_\tau$ corresponding to a tripod $\tau$ in G always lies in the interior of a separating 3-cycle C that contains all the vertices corresponding to its parent tripods on its boundary. To see this, observe that if $\tau$ has three parent $\tau _1$, $\tau _2$ and $\tau _3$, then C is formed by $v_{\tau _1}$, $v_{\tau _2}$ and $v_{\tau _3}$, such that $v_\tau$ is connected to each of them. If $\tau$ has two parent $\tau _1$ and $\tau _2$, then C is formed by $v_{\tau _1}$, $v_{\tau _2}$ and a subdivision vertex, such that $v_\tau$ is connected to $v_{\tau _1}$ and $v_{\tau _2}$. Finally, if $\tau$ has only one parent $\tau _1$, then C is formed by $v_{\tau _1}$ and two subdivision vertices, such that $v_\tau$ is connected to $v_{\tau _1}$. Since subdivision vertices are of degree 2, the result is a simple (possibly not maximal) planar 3-tree, which is a supergraph of H. To derive $H'$, we augment it to maximal by adding edges, while maintaining its embedding [23]. We are now ready to state the additional property that $H'$ has.

Lemma 4

Let $v_\tau$ and $v_{\tau _p}$ be two vertices of $H'$ that correspond to a tripod $\tau$ and to a parent tripod $\tau _p$ of $\tau$ in G. If $L_i$ and $L_j$ are the levels of $v_\tau$ and $v_{\tau _p}$ in the peeling-into-levels approach for $H'$, then $i\ge j$.

Proof

Let C be the inclusion-minimal separating 3-cycle of $H'$ containing $v_\tau$ in its interior and all vertices that correspond to the parent tripods of $\tau$ on its boundary. Let $L_k$, $L_l$ and $L_m$ be the levels of the three vertices of C, with $k \le l \le m$, in the peeling-into-levels approach for $H'$. As $\tau _p$ is a parent of $\tau$, $j \in \{k,l,m\}$ holds. Since C is a 3-cycle and since each edge in the peeling-into-level approach is either level or binding, $m\le k+1$ holds. The fact that $v_\tau$ lies in the interior of C and is connected to each of the vertices in $H'$ corresponding to its parent tripods in G, implies that $v_\tau$ is on level $L_{k+1}$, that is, $i=k+1$. So, $j \le m \le k+1=i$ holds. $\square$

As in the original algorithm by Dujmović et al. [5], we compute a 5-queue layout $QL (H')$ of $H'$. However, we use our modification of the algorithm by Alam et al. [10] described in Sect. 4.2. Denote by $x_1, \dots , x_h$ the vertices of the subgraph H of $H'$ as they appear in $QL (H')$, that is, we ignore subdivision vertices introduced when augmenting H to $H'$. Let also ${\mathcal {Q}}_0,\ldots ,{\mathcal {Q}}_4$ be the queues of $QL (H')$ as described in Invariants (T.1)–(T.4). To compute the linear layout $QL (G)$ of G, we use Lemma 1, which orders the vertices of each layer $V_i$ of the BFS-layering $\mathcal {L}$, with $i \ge 0$, as:

$$\begin{aligned} \overrightarrow{V_i}=A_{x_1}\cap V_i, A_{x_2}\cap V_i, \dots , A_{x_h}\cap V_i, \end{aligned}$$

where $A_{x_1},\ldots ,A_{x_h}$ are the bags, that is, the tripods, of the H-partition of G. Unlike in the original algorithm, we do not order the vertices in each subset $A_{x_j}\cap V_i$, with $j\in \{1,\ldots ,h\}$, arbitrarily. Instead, we carefully choose their order; we describe this choice in the rest of this section. Let $\tau$ be the tripod of bag $A_{x_j}$. Then, $A_{x_j}\cap V_i$ contains at most one vertex of each vertical path of $\tau$. We will order the three vertical paths of $\tau$, which defines the order of the (at most three) vertices of $A_{x_j}\cap V_i$ for every $i\ge 0$.

Let $L_l$ be the level of $v_\tau$ in the peeling-into-levels of $H'$, with $0 \le l< \lambda$, that is, we assume that $v_\tau$ is not a vertex of the last level $L_\lambda$ of $H'$. By Lemma 3, there are at most four connected components $c_s^1$, $c_s^2$, $c_t^1$ and $c_t^2$ of the subgraph $H_{l+1}'$ of $H'$ induced by the vertices of $L_{l+1}$, such that the edges connecting $v_\tau$ to vertices of $c_s^1$ and $c_s^2$ ($c_t^1$ and $c_t^2$) belong to ${\mathcal {Q}}_2$ (${\mathcal {Q}}_3$, respectively). Let c be one of $c_s^1$, $c_s^2$, $c_t^1$ and $c_t^2$; since c is a connected component of $H_{l+1}'$, it may contain vertices that correspond to tripods in G (that is, not subdivision vertices introduced while augmenting H to $H'$). We refer to the union of these vertices of G as the tripod-vertices of c, namely, the tripod-vertices of c are the vertices of G contained in the tripods of G that correspond to the vertices of c. By Invariant (T.1), $v_\tau$ precedes the vertices of $c_s^1$, $c_s^2$, $c_t^1$ and $c_t^2$ in $QL (H')$. Also by Invariant (T.2), we may assume that the vertices of $c_s^1$ ($c_t^1$) precede those of $c_s^2$ ($c_t^2$, respectively). Additionally, Lemma 3 ensures that the vertices of $c_s^2$ precede those of $c_t^1$.

Since an edge $(v_\tau ,v_{\tau '})$ exists in H, if $\tau$ is a parent of $\tau '$, or vice versa, by Lemma 4, for each vertex $v_{\tau '}$ of H that is a neighbor of $v_\tau$ in one of $c_s^1$, $c_s^2$, $c_t^1$ and $c_t^2$, it follows that $\tau$ is a parent of $\tau '$. By Property (P.6), there is a vertical path of $\tau$ in G, say p, such that no tripod-vertex of $c_s^2$ is adjacent to it in G. Similarly, there is a vertical path of $\tau$ in G, say q, such that no tripod-vertex of $c_t^2$ is adjacent to it in G. Note that p and q might be the same vertical path of $\tau$.

We now describe the order of the three vertical paths of $\tau$. We only specify the first one; the other two can be arbitrarily ordered:

• (i) if the tripod-vertices of $c_s^1$ and $c_s^2$ are connected to all three vertical paths of $\tau$ in G, then p is the first vertical path of $\tau$;

• (ii) if the tripod-vertices of $c_t^1$ and $c_t^2$ are connected to all three vertical paths of $\tau$ in G, then q is the first vertical path of $\tau$;

• (iii) otherwise, any vertical path of $\tau$ can be first.

Before proving formally that Cases (i) and (ii) cannot apply simultaneously (which implies that the order of the vertical paths of $\tau$ is well-defined), we state two important implications of choosing the first vertical path of $\tau$.

• (P.11) By assuming that all vertices of $c_s^1$ precede those of $c_s^2$, if tripod-vertices of $c_s^1$ and $c_s^2$ are connected to all three vertical paths of $\tau$, then tripod-vertices of $c_s^2$ are not connected to the first vertical path of $\tau$.

• (P.12) Also, under the assumption that all vertices of $c_t^1$ precede those of $c_t^2$, if tripod-vertices of $c_t^1$ and $c_t^2$ are connected to all three vertical paths of $\tau$, then tripod-vertices of $c_t^2$ are not connected to the first vertical path of $\tau$.

Fig. 8

Illustrations for: a–b the fact that Cases (i) and (ii) in the selection of the first vertical path of $\tau$ cannot apply simultaneously, and c the proof of Lemma 5

We now prove that Cases (i) and (ii) in the selection of the first vertical path of $\tau$ cannot apply simultaneously. Assume for a contradiction that the tripod-vertices of $c_s^1$ and $c_s^2$ are connected to all three vertical paths of $\tau$, and also that the tripod-vertices of $c_t^1$ and $c_t^2$ are connected to all three vertical paths of $\tau$. Since $v_\tau$ is the top vertex of the two triangular faces $f_t^1$ and $f_t^2$ that contain $c_t^1$ and $c_t^2$ in their interior in $\Gamma (H'_l)$, it follows that the edges connecting $v_\tau$ to $c_t^1$ and $c_t^2$ appear consecutively around $v_\tau$ in the cyclic order of the binding edges between levels $L_l$ and $L_{l+1}$ of $H'$; refer to the blue edges of Fig. 8a. Also, since $v_\tau$ is the middle vertex of the two triangular faces $f_s^1$ and $f_s^2$ that contain $c_s^1$ and $c_s^2$ in their interior in $\Gamma (H'_l)$, we can assume that the edges connecting $v_\tau$ to $c_s^1$ immediately precede the edges connecting $v_\tau$ to $c_t^1$ and $c_t^2$, while the edges connecting $v_\tau$ to $c_s^2$ immediately follow them; refer to the green edges of Fig. 8a. Note that since we have assumed that the vertices of $c_s^1$ precede the vertices of $c_s^2$, our assumption on the order of the edges around $v_\tau$ is a property of the algorithm by Alam et al. [10]. Now, if the tripod-vertices of $c_s^1$ and $c_s^2$ are connected to all three vertical paths of $\tau$, and the same holds for the tripod-vertices of $c_t^1$ and $c_t^2$, then Property (P.10) implies that in G the edges connecting vertices of $\tau$ to tripod-vertices of $c_t^1$ and $c_t^2$ alternate with edges connecting vertices of $\tau$ to tripod-vertices of $c_s^1$ and $c_s^2$; see e.g. Fig. 8b. Hence, the cyclic order of the edges around $v_\tau$ in $H'$ is not the same as the cyclic order of the edges incident to vertices of $\tau$ along a clockwise traversal of $\tau$ in G, contradicting Property (P.9). Hence, Case (i) and (ii) cannot apply simultaneously, as we initially claimed.

5 Reducing the Bound

To reduce the upper bound on the queue number of planar graphs, we turn our attention to the analysis of the required number of queues for the intra-bag inter-layer edges (E.2), as well as for the inter-bag edges (that is, either intra-layer (E.3) or inter-layer (E.4)–(E.5)); refer to Section 3.3.

For intra-bag inter-layer edges (E.2), the original algorithm by Dujmović et al. [5] uses three queues, since the layered width $\ell$ is 3; see blue edges in Fig. 4a. We prove that no three intra-bag inter-layer edges form a 3-rainbow, implying that the upper bound on the queue number of planar graphs can be improved from 49 to 48. Note that this first improvement does not require any of the modifications presented in Sect. 4.

Lemma 5

In the queue layout computed by the algorithm by Dujmović et al. [5], no three intra-bag inter-layer edges of G form a 3-rainbow.

Proof

Assume to the contrary that there exist three such edges $(u_1,v_1)$, $(u_2,v_2)$ and $(u_3,v_3)$ forming a 3-rainbow in $QL (G)$ such that $u_1 \prec _G u_2 \prec _G u_3 \prec _G v_3 \prec _G v_2 \prec _G v_1$. By Property (P.1) these edges belong to the same bag A of the H-partition, while by Property (P.2) their endpoints belong to two consecutive layers $V_i$ and $V_{i+1}$ of $\mathcal {L}$, for some $i\ge 0$. The underlying linear order implies that vertices $u_1$, $u_2$ and $u_3$ belong to $V_i$, while $v_1$, $v_2$ and $v_3$ belong to $V_{i+1}$. The order of $A\cap V_i$ and $A\cap V_{i+1}$ is $u_1 \prec _G u_2 \prec _G u_3$ and $v_3 \prec _G v_2 \prec _G v_1$; see Fig. 8c. Let $p_1$, $p_2$ and $p_3$ be the first, second and third vertical paths of tripod $\tau$ forming A. Then, $(u_1,v_3)\in p_1$, $(u_2,v_2)\in p_2$ and $(u_3,v_1)\in p_3$. However, $(u_1,v_1)$ and $(u_3,v_3)$ contradict Property (P.8). Hence, there is no 3-rainbow formed by intra-bag inter-layer edges of G. $\square$

For inter-bag edges (E.3)–(E.5), the algorithm by Dujmović et al. [5] uses $3\cdot 15$ queues, since the layered width $\ell$ is 3 and the best known upper bound on the queue number k of planar 3-trees is 5; see green, purple and orange edges in Fig. 4b. We exploit Property (P.3) to prove that $3\cdot 13$ queues suffice for inter-bag edges. This further improves the upper bound on the queue number of planar graphs from 48 to $42$. Note that this proof is more involved than the previous one and requires all modifications presented in Sect. 4.

Lemma 6

In the queue layout computed by our modification of the algorithm by Dujmović et al. [5], the inter-bag edges of G do not form a 40-rainbow.

Proof

To prove the statement, it suffices to show that the inter-bag edges of G that belong to each of (E.3), (E.4) and (E.5) form at most a 13-rainbow. We focus on the edges of (E.3), that is, on the intra-layer inter-bag edges of G; a similar argument applies for the other two types of edges, that is, forward and backward inter-layer inter-bag edges. We partition the edges of (E.3) into five parts $E_3^0,\ldots ,E_3^4$, such that for $i=0,\ldots ,4$, part $E_3^i$ contains the following type-(E.3) edges of G:

$$\begin{aligned} E_3^{i} =\{(u,v) \in \text{(E.3) }: u\in A_x, v\in A_y, (x,y) \in {\mathcal {Q}}_i \}. \end{aligned}$$

By Property (P.3), and since $\ell =3$, the edges of $E_3^i$ can form an r-rainbow with $r\le 3$ (this property actually implies the initial bound of $3\cdot 5 =15$ queues for all intra-layer inter-bag edges of G by Dujmović et al. [5]). We next prove that neither the edges of $E_3^2$ nor the edges of $E_3^3$ form a 3-rainbow, which yields the desired reduction from 15 to 13 on the size of the maximum rainbow formed by all edges of (E.3). Assume that this is not true, say for the edges of $E_3^2$; a symmetric argument applies for the edges of $E_3^3$. Let $(u_1,v_1)$, $(u_2,v_2)$ and $(u_3,v_3)$ be three edge of $E_3^2$ that form a 3-rainbow, such that $u_1\prec _G u_2 \prec _G u_3 \prec _G v_3\prec _G v_2\prec _G v_1$. Since $(u_1,v_1)$, $(u_2,v_2)$ and $(u_3,v_3)$ are of type (E.3), that is, intra-layer edges, their endpoints $u_1$, $u_2$, $u_3$, $v_1$, $v_2$ and $v_3$ all belong to the same layer of the BFS-layering $\mathcal {L}$ of G⁵. Assuming that $u_i \in A_{x_i}$ and $v_i \in A_{y_i}$, it follows that $(x_i,y_i)$ is an edge of H assigned to ${\mathcal {Q}}_2$ and $x_i\prec _H y_i$, for $i=1,2,3$.

By Property (P.3), either $x_1=x_2=x_3$ or $y_1=y_2=y_3$ holds. First, we prove that $x_1=x_2=x_3$ always holds. Assume for a contradiction that, without loss of generality, $x_1\ne x_2$. Then, $y=y_1=y_2=y_3$ holds. Since the edges of ${\mathcal {Q}}_2$ are binding in the peeling-into-levels decomposition of $H'$ and since $x_1\prec _H x_2\prec _H y$, vertices $x_1$ and $x_2$ belong to level $L_j$ of $H'$, while y belongs to level $L_{j+1}$, for some $0\le j<\lambda$. By Property (T.4), y has only one neighbor on level $L_j$ such that the edge connecting y to this neighbor is assigned to ${\mathcal {Q}}_2$. This contradicts the fact that both edges $(x_1,y)$ and $(x_2,y)$ are assigned to ${\mathcal {Q}}_2$, since $x_1 \ne x_2$ and $x_1$ and $x_2$ belong to $L_j$. Hence $x=x_1=x_2=x_3$ holds.

Now, since the edges of ${\mathcal {Q}}_2$ are binding edges in the peeling-into-levels decomposition of $H'$, and since, for $i=1,2,3$ , $x \prec _H y_i$, it follows that vertex x belongs to level $L_j$ of $H'$, while vertex $y_i$ belongs to level $L_{j+1}$, for some $0\le j<\lambda$. Let $\tau$ be the tripod of bag $A_x$ and let $\tau _1$, $\tau _2$, $\tau _3$ be the tripods of $A_{y_1}$, $A_{y_2}$ and $A_{y_3}$, respectively. Note that vertices $y_1$, $y_2$ and $y_3$ are not necessarily distinct, that is, the corresponding tripods $\tau _1$, $\tau _2$ and $\tau _3$ are not necessarily pairwise different. However, since $u_1\prec _G u_2\prec _G u_3$, and since $u_1$, $u_2$ and $u_3$ belong to the same layer in the BFS-layering $\mathcal {L}$ of G, we can conclude that $u_1$, $u_2$ and $u_3$ belong to $p_1$, $p_2$ and $p_3$, respectively, where $p_1$, $p_2$ and $p_3$ are the first, second and third vertical paths of $\tau$. From the fact that, for $i=1,2,3$, the edge $(x,y_i)$ belongs to H, we have that $\tau$ is a parent tripod of $\tau _i$ or vice versa. Since x belongs to level $L_j$ of $H'$ and vertices $y_1$, $y_2$ and $y_3$ belong to level $L_{j+1}$, Lemma 4 implies that $\tau$ is a parent tripod of $\tau _1$, $\tau _2$ and $\tau _3$. By Lemma 3, there exist at most two connected components $c_s^1$ and $c_s^2$ of $H'_{j+1}$, such that the edges connecting x to vertices of $c_s^1$ and $c_s^2$ have been assigned to ${\mathcal {Q}}_2$. Without loss of generality, by Invariant (T.2) we assume that all vertices of $c_s^1$ (if any) precede those of $c_s^2$ (if any).

We first argue that not all vertices $y_1$, $y_2$ and $y_3$ belong to only one of $c_s^1$ or $c_s^2$, which further implies that both components $c_s^1$ and $c_s^2$ exist. Assume to the contrary that $y_1$, $y_2$ and $y_3$ belong to $c_s^1$; a similar argument applies for $c_s^2$. By Property (P.6), tripod-vertices of component $c_s^1$ are not adjacent to the vertices of at least one vertical path $p_i$ of $\tau$ in G, where $1\le i\le 3$. In this case, however, we obtain a contradiction to the fact that $v_i$, which is a tripod-vertex of $c_s^1$, is connect to $u_i$ that belongs to $p_i$, i.e., edge $(u_i,v_i)$ cannot exist in G.

Hence, not all vertices $y_1$, $y_2$ and $y_3$ belong to only one of $c_s^1$ or $c_s^2$, as we initially claimed. In particular, since $y_3 \preceq _H y_2 \preceq _H y_1$ and since all vertices of $c_s^1$ precede those of $c_s^2$, it follows that $y_3$ belongs to $c_s^1$, while $y_1$ belongs to $c_s^2$. On the other hand, vertices $u_1$, $u_2$ and $u_3$ belong to $p_1$, $p_2$ and $p_3$, respectively, which implies that the vertices of $c_s^1$ and $c_s^2$ are connected to all vertical paths of $\tau$. Thus, Property (P.11) applies, which states that the tripod-vertices of $c_s^2$ (including $v_1$) are not connected to the first vertical path $p_1$ of $\tau$. However, this contradicts the fact that $(u_1,v_1)$ is an edge of G, with $u_1\in p_1$. Hence, the edges of $E_3^2$ cannot form a 3-rainbow. Using Property (P.12) instead of Property (P.11), we can symmetrically prove that the edges of $E_3^3$ cannot form a 3-rainbow. This completes the proof of the lemma. $\square$

We are now ready to state the main theorem of this section by combining Lemmas 5 and 6.

Theorem 7

Every planar graph has queue number at most $42$.

6 Conclusions

In this work, we improved the upper bound on the queue number of planar graphs by refining the three techniques involved in the original algorithm [5]. We believe that our approach has the potential to further reduce the upper bound by at least 3 more queues (i.e., from $42$ to 39). However, more elaborate arguments that exploit planarity in more depth are required, and several details have to be worked out. Towards narrowing the gap with the lower bound of 4, determining the exact queue number of planar 3-trees becomes critical; an improvement of the current upper bound of 5 (to meet the lower bound of 4) implies a direct improvement on the upper bound on the queue number of general planar graphs. On the other hand, to obtain a better understanding of the general problem, it is also reasonable to further examine subclasses of planar graphs, such as bipartite planar graphs or planar graphs with bounded degree (e.g., cubic planar graphs).