A Universal Point Set for 2-Outerplanar Graphs

Abstract

A point set $S \subseteq \mathbb {R}^2$ is universal for a class $\mathcal G$ if every graph of $\mathcal{G}$ has a planar straight-line embedding on S. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a sub-quadratic universal point set for them is one of the most fascinating open problems in Graph Drawing. Motivated by the fact that outerplanarity is a key property for the existence of small universal point sets, we study 2-outerplanar graphs and provide for them a universal point set of size $O(n \log n)$.

This work has been supported by DFG grant Ka812/17-1. The full version of the paper, including all the missing proofs, can be found in [2].

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1 Introduction

Let S be a set of m points on the plane. A planar straight-line embedding of an n-vertex planar graph G, with $n\le m$, on S is a mapping of each vertex of G to a distinct point of S so that, if the edges are drawn straight-line, no two edges cross. Point set S is universal for a class $\mathcal G$ of graphs if every graph $G \in \mathcal{G}$ has a planar straight-line embedding on S. Asymptotically, the smallest universal point set for general planar graphs is known to have size at least 1.235n [11], while the upper bound is $O(n^2)$ [3, 8, 12]. All the upper bounds are based on drawing the graphs on an integer grid, except for the one by Bannister et al. [3], who use super-patterns to obtain a universal point set of size $n^2/4-\varTheta (n)$ – currently the best result for planar graphs. Closing the gap between the lower and the upper bounds is a challenging open problem [6–8].

A subclass of planar graphs for which the “smallest possible” universal point set is known is the class of outerplanar graphs – the graphs that admit a straight-line planar drawing in which all vertices are incident to the outer face. Namely, Gritzmann et al. [10] and Bose [5] proved that any size-n point set in general position is universal for n-vertex outerplanar graphs. Motivated by this result, we consider the class of k-outerplanar graphs, with $k \ge 2$, which is a generalization of outerplanar graphs. A planar drawing of a graph is k-outerplanar if removing the vertices of the outer face, called k-th level, produces a $(k-1)$-outerplanar drawing, where 1-outerplanar stands for outerplanar. A graph is k-outerplanar if it admits a k-outerplanar drawing. Note that every planar graph is a k-outerplanar graph, for some value of $k \in O(n)$. Hence, in order to tackle a meaningful subproblem of the general one, it makes sense to study the existence of subquadratic universal point sets when the value of k is bounded by a constant or a sublinear function. However, while the case $k=1$ is trivially solved by selecting any n points in general position, as observed above [5, 10], the case $k=2$ already eluded several attempts of solution and turned out to be far from trivial. In this paper, we finally solve the case $k=2$ by providing a universal point set for 2-outerplanar graphs of size $O(n \log n)$.

A subclass of k-outerplanar graphs, in which the value of k is unbounded, but every level is restricted to be a chordless simple cycle, was known to have a universal point set of size $O(n (\frac{\log n}{\log \log n})^2)$ [1], which was subsequently reduced to $O(n \log n)$ [3]. It is also known that planar 3-trees – graphs not defined in terms of k-outerplanarity – have a universal point set of size $O(n^{5/3})$ [9]. Note that planar 3-trees have treewidth equal to 3, while 2-outerplanar graphs have treewidth at most 5.

Structure of the Paper: After some preliminaries and definitions in Sect. 2, we consider 2-outerplanar graphs in Sect. 3 where the inner level is a forest and all the internal faces are triangles. We prove that this class of graphs admits a universal point set of size $O(n^{3/2})$. We then extend the result in Sect. 4 to 2-outerplanar graphs in which the inner level is still a forest but the faces are allowed to have larger size. Finally, in Sect. 5, we outline how the result of Sect. 4 can be extended to general 2-outerplanar graphs. We also explain how to apply the methods in [3] to reduce the size of the point set to $O(n \log n)$. We conclude with open problems in Sect. 6.

2 Preliminaries and Definitions

A straight-line segment with endpoints p and q is denoted by s(pq). A circular arc with endpoints p and q (clockwise) is denoted by a(pq). We assume familiarity with the concepts of planar graphs, straight-line planar drawings and their faces. A straight-line planar drawing $\varGamma$ of a graph G determines a clockwise ordering of the edges incident to each vertex u of G, called rotation at u. The rotation scheme of G in $\varGamma$ is the set of the rotations at all the vertices of G determined by $\varGamma$. Observe that, if G is connected, in all the straight-line planar drawings of G determining the same rotation scheme, the faces of the drawing are delimited by the same edges.

Let $[G,\mathcal{H}]$ be a 2-outerplanar graph, where the outer level is an outerplanar graph G and the inner level is a set $\mathcal{H} = \{G_1,\dots ,G_k\}$ of outerplanar graphs. We assume that $[G,\mathcal{H}]$ is given together with a rotation scheme, and the goal is to construct a planar straight-line embedding of $[G,\mathcal{H}]$ on a point set determining this rotation scheme. Since $[G,\mathcal{H}]$ can be assumed to be connected (as otherwise we can add a minimal set of dummy edges to make it connected), this is equivalent to assuming that a straight-line planar drawing $\varGamma$ of $[G,\mathcal{H}]$ is given. We rename the faces of $\varGamma$ as $F_1,\dots ,F_k$ in such a way that each graph $G_h$, which can also be assumed connected, lies inside face $F_h$. Note that, for each face $F_h$ of G, the graph $[F_h,G_h]$ is again a 2-outerplanar graph; however, its outer level $F_h$ is a simple chordless cycle and its inner level $G_h$ consists of only one connected component. In the special case in which $G_h$ is a tree we say that graph $[F_h,G_h]$ is a cycle-tree graph. We say that a 2-outerplanar graph is inner-triangulated if all the internal faces are 3-cycles. Note that not every cycle-tree graph can be augmented to be inner-triangulated without introducing multiple edges.

Fig. 1.

(a) Illustration of S, focused on $S_j$ of $p_j$. (b) A cycle-tree graph and its embedding.

3 Inner-Triangulated 2-Outerplanar Graphs with Forest

In this section we prove that there exists a universal point set S of size $O(n^{3/2})$ for the class of n-vertex inner-triangulated 2-outerplanar graphs $[G,\mathcal{H}]$ where $\mathcal{H}$ is a forest.

3.1 Construction of the Universal Point Set

In the following we describe S (Fig. 1(a)). Let $\pi$ be a half circle with center O and let $N:=n+\sqrt{n}$. Uniformly distribute points in $S_\mathcal{M}=\{p_1,\dots ,p_N\}$ on $\pi$. The points in $S_\mathcal{D}=\{p_{i\sqrt{n}+i}: 1\le i \le \sqrt{n} \}$ are called dense, while the remaining points in $S_\mathcal{M} \setminus S_\mathcal{D}$ are sparse ¹. For $j = 2, \dots , N-1$, place a circle $\pi _j$ with its center $p_j^C$ on $s(p_jO)$, so that it lies completely inside the triangle $\triangle p_{j-1}p_jp_{j+1}$ and inside the triangle $\triangle p_1p_jp_N$. Note that the angles $\angle p_jp_j^Cp_N$ and $\angle p_jp_j^Cp_1$ are smaller than $180^\circ$. Let $p_j^N$ be the intersection point between $s(p_jO)$ and $\pi _j$ that is closer to O. Also, let $p_j^1$ (resp. $p_j^2$) be the intersection point of $s(p_j^Cp_{j+1})$ (resp. $s(p_j^Cp_{j-1})$) with $\pi _j$. Finally, let $p_j^3$ (resp. $p_j^4$) be the intersection point of $\pi _j$ with its diameter orthogonal to $s(p_jO)$, such that $a(p_j^3p_j^4)$ does not contain $p_j^N$. Now, choose a point $p_j^+$ on the arc $a(p_j^1p_j^3)$, and a point $p_j^-$ on the arc $a(p_j^4p_j^2)$. To complete the construction of S, evenly distribute $\overline{n}-1$ points on each of the three segments $s^N_j := s(p_j^Cp_j^N)$, $s^+_j := s(p_j^Cp_j^+)$, and $s^-_j := s(p_j^Cp_j^-)$, where $\overline{n} = n$ if $p_j$ is dense and $\overline{n} = \sqrt{n}$ if it is sparse. We refer to the points on $s^N, s^+, s^-$, including the points $p_j^N, p_j^C, p_j^+, p_j^-$, as the point set of $p_j$, and we denote it by $S_j$. Vertex $p_j^C$ is the center vertex of $S_j$. The described construction uses $O(n^{3/2})$ points and ensures the following property.

Property 1

For each $j = 1, \dots , N$, the following visibility properties hold:

(A) The straight-line segments connecting point $p_j$ to: point $p_j^-$, to the points on $s^-_j$, to $p_j^C$, to the points on $s^+_j$, and to $p_j^+$ appear in this clockwise order around $p_j$.

(B) For all $l < j$, consider any point $x_l \in \{p_l\} \cup S_l$ (see Fig. 1); then, the straight-line segments connecting $x_l$ to: $p_j^N$, to the points on $s^N_j$, to $p_j^C$, to the points on $s^-_j$, to $p_j^-$, and to $p_j$ appear in this clockwise order around $x_l$. Also, consider the line passing through $x_l$ and any point in $\{p_j\} \cup S_j$; then, every point in $\{p_q\} \cup S_q$, with $l < q < j$, lies in the half-plane delimited by this line that does not contain the center O of $\pi$.

(C) For all $l > j$, consider any point $x_l \in \{p_l\} \cup S_l$; then, the straight-line segments connecting $x_l$ to: $p_j^N$, to the points on $s^N_j$, to $p_j^C$, to the points on $s^+_j$, to $p_j^+$, and to $p_j$ appear in this counterclockwise order around $x_l$. Also, consider the line passing through $x_l$ and any point in $\{p_j\} \cup S_j$; then, every point in $\{p_q\} \cup S_q$, with $j < q < l$, lies in the half-plane delimited by this line that does not contain O.

3.2 Labeling the Graph

Let $[G,\mathcal{H}]$ be an inner-triangulated 2-outerplanar graph where G is an outerplanar graph and $\mathcal{H}=\{T_1,\dots ,T_k\}$ is a forest such that tree $T_h$ lies inside face $F_h$ of G, for each $1 \le h \le k$. The idea behind the labeling is the following: in our embedding strategy, G will be embedded on the half-circle $\pi$ of the point set S, while the tree $T_h \in \mathcal{H}$ lying inside each face $F_h$ of G will be embedded on the point sets $S_j$ of some of the points $p_j$ on which vertices of $F_h$ are placed. Note that, since $\pi$ is a half-circle, the drawing of $F_h$ will always be a convex polygon in which two vertices have small (acute) internal angles, while all the other vertices have large (obtuse) internal angles. In particular, the vertices with the small angle are the first and the last vertices of $F_h$ in the order in which they appear along the outer face of $\varGamma$. Since, by construction, a point $p_j$ of $F_h$ has its point set $S_j$ in the interior of $F_h$ if and only if it has a large angle, we aim at assigning each vertex of $T_h$ to a vertex of $F_h$ that is neither the first nor the last. We will describe this assignment by means of a labeling $\ell :[G,\mathcal{H}] \rightarrow 1, \dots , |G|$; namely, we will assign a distinct label $\ell (v)$ to each vertex $v \in G$ and then assign to each vertex of $T_h$ the same label as one of the vertices of $F_h$ that is neither the first or the last. Then, the number of vertices with the same label as a vertex of G will determine whether this vertex will be placed on a sparse or a dense point. We formalize this idea in the following.

We rename the vertices of G as $v_1, \dots , v_{|G|}$ in the order in which they appear along the outer face of $\varGamma$, and label them with $\ell (v_i)=i$ for $i=1,\dots ,|G|$. Next, we label the vertices of each tree $T_h \in \mathcal{H}$. Since trees $T_h$ and $T_{h'}$ are disjoint for $h \ne h'$, we focus on the cycle-tree graph [F, T] composed of a single face $F=F_h$ of G and of the tree $T=T_h \in \mathcal{H}$ inside it. Rename the vertices of F as $w_1, \dots , w_m$ in such a way that for any two vertices $w_x=v_p$ and $w_{x+1}=v_q$, where $p,q \in \{1,\dots ,|G|\}$, it holds that $p < q$. As a result, $w_1$ and $w_m$ are the only vertices of F with small internal angles. A vertex of T is a fork vertex if it is adjacent to more than two vertices of F (square vertices in Fig. 1(b)), otherwise it is a non-fork vertex (cross vertices in Fig. 1(b)). Since [F, T] is inner-triangulated, every vertex of T is adjacent to at least two vertices of F, and hence non-fork vertices are adjacent to exactly two vertices of F. We label the vertices of T starting from its fork vertices. To this end, we construct a tree $T'$ composed only of the fork vertices, as follows. Initialize $T'$=T. Then, as long as there exists a non-fork vertex of degree 3 (namely, with 2 neighbors in F and 1 in $T'$), remove it and its incident edges from $T'$. The vertices removed in this step are called foliage (small crosses in Fig. 1(b)). All the remaining non-fork vertices have degree 4 (namely 2 in F and 2 in $T'$); for each of them, remove it and its incident edges from $T'$ and add an edge between the two vertices of $T'$ that were connected to it before its removal. The vertices removed in this step are branch vertices (large crosses in Fig. 1(b)). A vertex $w_x \in F$ is called free if so far no vertex of $T'$ has label $\ell (w_x)$. To perform the labeling, we traverse $T'$ bottom-up with respect to a root r that is the vertex of $T'$ adjacent to both $w_1$ and $w_m$. Since [F, T] is inner-triangulated, this vertex is unique. During the traversal of $T'$, we maintain the invariant that vertices of $T'$ are incident to only free vertices of F. Initially the invariant is satisfied since all the vertices of F are free. Let a be the fork vertex considered in a step of the traversal of $T'$, and let $w_{a_1},\dots ,w_{a_k}$ be the vertices of F adjacent to a, with $1 \le a_1 <\dots <a_k \le m$ and $k \ge 3$. By the invariant, $w_{a_1},\dots ,w_{a_k}$ are free. Choose any vertex $w_{a_i}$ such that $2 \le i \le k-1$, and set $\ell (a)=\ell (w_{a_i})$. For example, the red fork vertex in Fig. 1(b) adjacent to $w_3$, $w_4$, and $w_5$ in F gets label $\ell (w_4)$. Since vertices $w_{a_2},\dots ,w_{a_{k-1}}$ cannot be adjacent to any vertex of $T'$ that is visited after a in the bottom-up traversal, the invariant is maintained at the end of each step. When finally a=r, then $w_{a_1}=w_1$ and $w_{a_k}=w_m$ are both free.

Now we label the non-fork vertices of T based on the labeling of $T'$. Let b be a non-fork vertex. If b is a branch vertex, then consider the first fork vertex a encountered on a path from b to a leaf of T; set $\ell (b)=\ell (a)$. Otherwise, b is a foliage vertex. In this case, consider the first fork vertex $a'$ encountered on a path from b to the root r of T. Let $v,w \in F$ be the two vertices of F adjacent to b; assume $\ell (v) < \ell (w)$. If $\ell (a') \le \ell (v)$, then set $\ell (b)=\ell (v)$; if $\ell (a') \ge \ell (w)$, then set $\ell (b)=\ell (w)$; and if $\ell (v) < \ell (a') < \ell (w)$, then set $\ell (b)=\ell (a')$ (the latter case only happens when $a'$ is the root and b is adjacent to $w_1$ and $w_m$). Note that the described algorithm ensures that adjacent non-fork vertices have the same label. We perform the labeling procedure for every $T_h \in \mathcal{H}$ and obtain a labeling for $[G,\mathcal{H}]$. We say that the subgraph of $\mathcal{H}$ induced by all the vertices of $\mathcal{H}$ with label i is the restricted subgraph $H_i$ of $\mathcal{H}$ for all $i= 1, \dots , |G|$ (see Fig. 1(b)).

Lemma 1

Each restricted subgraph $H_i$ of $\mathcal H$, $1 \le i \le |G|$, is a tree all of whose vertices have degree at most 2, except for one vertex that may have degree 3.

Proof Sketch

First, $H_i$ has at most one fork vertex a, which is hence the only one with degree larger than 2. Further, a is incident to at most one path (to no path, if $a=r$) of branch vertices, namely the one connecting it to its parent fork vertex. Finally, a is incident to at most two (if $a \ne r$) or at most three (if $a=r$) paths of foliage vertices, namely the ones whose vertices are incident to the vertex $w \in F$ such that $\ell (w)=i$. $\square$

3.3 Embedding on the Point Set

We describe an embedding algorithm consisting of three steps (see Fig. 1(b)).

Step a: Let $\omega : G \rightarrow \mathbb {N}$ be a weight function with $\omega (v_i)$=$|\{v \in [G,\mathcal{H}] \mid \ell (v) = i\}|$ for every $v_i \in G$. Note that $\sum _{v_i \in G}{\omega (v_i)} = n$. We categorize each vertex $v_i \in G$ as sparse if $1\le \omega (v_i) \le \sqrt{n}$, and dense if $\omega (v_i) > \sqrt{n}$. There are at most $\sqrt{n}$ dense vertices.

Step b: We draw the vertices $v_1,\dots ,v_{|G|}$ of G on the $N := n + \sqrt{n}$ points of $\pi$ in the same order as they appear along the outer face of $\varGamma$, in such a way that dense (resp. sparse) vertices are placed on dense (resp. sparse) points. The resulting embedding $\widetilde{\varGamma }$ of G is planar since $\varGamma$ is planar. The construction of $\widetilde{\varGamma }$ implies the following.

Property 2

Let $Q=\{p_{j_1}, \dots , p_{j_m}\} \subseteq \pi$, $j_i < j_{i+1}$, be the polygon representing a face of G. Polygon Q contains in its interior all the point sets $S_{j_2}, \dots , S_{j_{m-1}}$.

Step c: Finally, we consider forest $\mathcal{H} = \{T_1,\dots ,T_k\}$. We describe the embedding algorithm for a single cycle-tree graph [F, T], where $F = w_1, \dots , w_m$ is a face of G and $T \in \mathcal{H}$ is the tree lying inside F. We show how to embed the restricted subgraph $H_i$, for each vertex $w_x$ of F with label $\ell (w_x)=i$, on the point set $S_j$ of the point $p_j$ where $w_x$ is placed. We remark that the labeling procedure ensures that $|H_i|+1 = \omega (w_x) \le |S_j|$; also, by Property 2, point set $S_j$ lies inside the polygon representing F, except for the two points where vertices $w_1$ and $w_m$ have been placed.

By Lemma 1, $H_i$ has at most one (fork-)vertex a of degree 3, while all other vertices have smaller degree. We place a, if any, on the center point $p_j^C$ of $p_j$. The at most three paths of non-fork vertices are placed on segments $s^+_j, s^-_j, s^N_j$ starting from $p_j^C$; namely, the unique path of branch vertices is placed on $s^N_j$, while the two paths of foliage vertices are placed on $s^+_j$ or $s^-_j$ based on whether the vertex of G different from $w_x$ they are incident to is $w_{x+1}$ or $w_{x-1}$, respectively. If $a=r$, then the path of foliage vertices incident to $w_1$ and $w_m$ is placed on $s^N_j$.

Fig. 2.

(a)(top) P contains $a' \ne a$, (a)(bottom) $a'$ is a leaf of T. (b)–(c) Dark-gray triangles are used for construction of petal points $r(p^{+}_z)$ while light-gray triangles for $l(p^{+}_z)$.

We show that this results in a planar drawing of T. First, for every two fork vertices $a \in H_p$ and $a' \in H_q$, with $p < q$, all the leaves of the subtree of T rooted at a have smaller label than all the leaves of the subtree of T rooted at $a'$. Then, for each $w_x \in F$, with $\ell (w_x)=i$, consider the fork vertex $a \in H_i$, which lies on $p_j^C$. Let P be any path connecting a to a leaf of T and let $a^*$ be the neighbor of a in P. If P contains a fork vertex other than a (Fig. 2(a)), then let $a'$ be the fork vertex in P that is closest to a (possibly $a'$=$a^*$) and let $p_q^C$ be the point where $a'$ has been placed. Assume $q < j$, the case $q > j$ is analogous. By definition, the non-fork vertices in the path from a to $a'$ (if any) are branch vertices, and hence lie on $s_q^N$. Then, Property 1 ensures that the straight-line edge $(a,a^*)$ separates all the point sets $S_p$ with $q < p < j$ from the center of $\pi$. Since the vertices on $S_p$ are only connected either to each other or to the vertices on $s_j^-$ and $s_q^+$, edge $(a,a^*)$ is not involved in any crossing. If P does not contain any fork vertex other than a (Fig. 2(a)), then all the vertices of P other than a are foliage vertices and are placed on a segment $s_q^+$ or $s_q^-$, for some q. In particular, if $q < j$, then they are on $s_q^-$; if $q > j$, then they are on $s_q^+$; while if $q = j$, then they are either on $s_q^+$ or on $s_q^-$. In all the cases, Property 1 ensures that edge $(a,a^*)$ does not cross any edge.

Finally, observe that any path of T containing only non-fork vertices is placed on the same segment of the point set, and hence its edges do not cross. As for the edges connecting vertices in one of these paths to the two leaves of T they are connected to, note that by item (A) of Property 1 the edges between each of these leaves and these vertices appear in the rotation at the leaf in the same order as they appear in the path.

Lemma 2

There exists a universal point set of size $O(n^{3/2})$ for the class of n-vertex inner-triangulated 2-outerplanar graphs $[G,\mathcal{H}]$ where $\mathcal{H}$ is a forest.

Fig. 3.

(a)–(c): Insertion of triangulation edges in (a) a petal face, (b) a non-protected big face, and (c) a big face protected by vertex $b_1$. (d)–(e) Illustration of the two cases for removing bad faces. Face g is petal in (d) and big in (e). Dummy edges are dashed, the removed edge e is red (Color figure online).

4 2-Outerplanar Graphs with Forest

In this section we consider 2-outerplanar graphs $[G,\mathcal{H}]$ where $\mathcal{H}$ is a forest. Contrary to the previous section, we do not assume $[G,\mathcal{H}]$ to be inner-triangulated. As observed before, augmenting it might be not possible without introducing multiple edges. The main idea to overcome this problem is to first identify the parts of $[G,\mathcal{H}]$ not allowing for the augmentation, remove them, and augment the resulting graph with dummy edges to inner-triangulated (Sect. 4.2); then, apply Lemma 2 to embed the inner-triangulated graph on the point set S; and finally remove the dummy edges and embed the parts of the graph that had been previously removed on the remaining points (Sect. 4.3). To do so, we first need to extend the point set S with some additional points.

4.1 Extending the Universal Point Set

We construct a point set $S^*$ with $O(n^{3/2})$ points from S by adding petal points to segments $s^+_j, s^N_j, s^-_j$ of the point sets $S_j$, for every j=$2,\dots ,N-1$. For simplicity of notation, we skip the subscript j whenever possible. We denote by $p^\sigma _z$ the z-th point on segment $s^\sigma$, with $\sigma \in \{+,-,N\}$ and z=$1, \dots , \overline{n}$ (where $\overline{n}$=$\sqrt{n}$ or $\overline{n}$=n, depending on whether $p_j$ is sparse or dense), so that $p^\sigma _1$ is the point following $p^C$ along $s^\sigma$ and $p^\sigma _{\overline{n}} = p^\sigma _j$. For each point $p^\sigma _z$ we add two petal points $l(p^{\sigma }_z)$ and $r(p^{\sigma }_z)$ to $S^*$.

We first describe the procedure for $s^+$, see Fig. 2(c). For each z=$1, \dots , \overline{n}$, consider the intersection point $q_z$ between segments $s(p^+_{z-1} p_{j+1})$ and $s(p^+_{z} p_N)$, where $p^+_{z-1} = p_j^C$ when $z=1$. By construction, all triangles $\triangle p^+_{z-1} p^+_{z} q_z$ have two corners on $s^+$, have the other corner in the same half-plane delimited by the line through $s^+$, and do not intersect each other except at common corners. Hence, there exists a convex arc $\pi ^+_r$ passing through $p^C_j$ and $p^+_{\overline{n}}=p^+_j$, and intersecting the interior of every triangle. For each $z = 1, \dots , \overline{n}$, we place the petal point $r(p^{+}_z)$ on the arc of $\pi ^+_r$ lying inside triangle $\triangle p^+_{z-1} p^+_{z} q_z$. For the other petal point $l(p^{+}_z)$ we use the same procedure by considering triangles $\triangle p^+_{z-1} p^+_z p_j$ instead of $\triangle p^+_{z-1} p^+_z q_z$. Symmetrically we place the petal points for $s^-$, using points $p_{j-1}$ and $p_1$ to place $l(p^{-}_z)$ and point $p_j$ to place $r(p^{-}_z)$, and for $s^N$, using points $p_{j-1}$ and $p_1$ to place $l(p^{N}_z)$ and points $p_{j+1}$ and $p_N$ to place $r(p^{N}_z)$.

4.2 Modifying and Labeling the Graph

We now aim at modifying $[G,\mathcal{H}]$ to obtain an inner-triangulated graph that can be embedded on the original point set S (Part A and Part B); in Sect. 4.3 we describe how to exploit this embedding on S to obtain an embedding of the original graph $[G,\mathcal{H}]$ on the extended point set $S^*$ (Part C). We describe the procedure just for a cycle-tree graph [F, T] composed of a face F of G and of the tree T inside it.

Part A: We categorize each face f of [F, T] based on the number of vertices of F and of T that are incident to it. Since T is a tree, f has at least a vertex of F and a vertex of T incident to it. If f contains exactly one vertex of F, then it is a petal face. If f contains exactly one vertex of T, then it is a small face. Otherwise, it is a big face. Let $b_1,\dots ,b_l$ be the occurrences of the vertices of T in a clockwise order walk along the boundary of a big face f. If either $b_1$ or $b_l$, say $b_1$, has more than one adjacent vertex in F (namely one in f and at least one not in f), then f is protected by $b_1$. If f is a big face with exactly two vertices incident to F and is not protected, then f is a bad face.

The next lemma gives sufficient conditions to triangulate G without introducing multiple edges; we will later use this lemma to identify the “tree components” of T whose removal allows for a triangulation.

Lemma 3

Let [F, T] be a biconnected simple cycle-tree graph, such that (1) each vertex of F has degree at most four, and (2) there exists no bad face in [F, T]. It is possible to augment [F, T] to an inner-triangulated simple cycle-tree graph.

Proof Sketch

Each petal (small, respectively) face f can be triangulated by adding vertices between the only vertex of F (of T) incident to f and all the other vertices of f. Multiple edges are not created since [F, T] is biconnected and there exists no two petal faces incident to the same vertex v of F, as v has degree at most 4; see Fig. 3(a).

Consider a big face f, with vertex occurrences $v_1,...,v_{l'},b_1,...,b_l$ (with $l,l'>1$), where $v_1,...,v_{l'} \in F$ and $b_1,...,b_l \in T$. If f is protected by a vertex, say $b_1$, then it is triangulated by adding an edge between $b_l$ and every vertex of F, and an edge between $v_{l'}$ and every vertex of T; see Fig. 3(b). The absence of multiple edges is due to the edge connecting $b_1$ to a vertex of F not incident to f, which implies that $v_{l'}$ is not connected to any vertex of T incident to f other than $b_1$. Finally, if f is not protected by any vertex, we make it protected by adding an edge $(b_l,v_2)$ and apply the previous case; see Fig. 3(c). Since f is not a bad face, we have $l' > 2$, and hence $v_2$ is not connected to any vertex of T, which implies that $(b_l,v_2)$ is not a multiple edge.       $\square$

We now describe a procedure to transform cycle-tree graph [F, T] into another one $[F,T'']$ that is biconnected and satisfies the conditions of Lemma 3. We do this in two steps: first, we remove some edges connecting a vertex of F and a vertex of T to transform [F, T] into a cycle-tree graph $[F,T'$=T] that is not biconnected but that satisfies the two conditions; then, we remove the “tree components” of $T'$ that are not connected to vertices of F in order to obtain a cycle-tree graph $[F,T'' \subseteq T']$ that is also biconnected.

To satisfy condition (1) of Lemma 3, we merge all the petal faces incident to the same vertex of F into a single one by repeatedly removing an edge shared by two adjacent petal faces. We refer to these removed edges as petal edges, denoted by $E_P$.

To satisfy condition (2) of Lemma 3, we consider each bad face $f = v_1,v_2,b_1,\dots ,b_l$, where $v_1,v_2 \in F$ and $b_1,\dots ,b_l \in T$. Let g be the face incident to $v_1$ sharing edge $e=(v_1,b_l)$ with f. We remove e, hence merging f and g into a single face $f'$, that we split again by adding dummy edges, based on the type of face g, in such a way that no new bad face is created. Since f is a bad face, it is not protected by $b_l$, and hence g is not a small face. If g is a petal face, then $f'$ is still a big face with two vertices of F incident to it, namely $v_1$ and $v_2$; see Fig. 3(d). We add edge $(v_1,b_1)$, splitting $f'$ into a petal face $v_1,b_1,\dots ,b_l$ and a triangular face $v_1,v_2,b_1$. If g is a big face, then $f'$ is a big face; see Fig. 3(e). Let $g=w_1,\dots ,w_q,c_1,\dots ,c_h$, where $w_1,\dots ,w_q \in F$, with $w_q=v_1$, and $c_1,\dots ,c_h \in T$, with $c_1=b_l$. We add two dummy edges $(v_1,c_h)$ and $(v_1,b_1)$, splitting $f'$ into a small face $w_1,\dots ,w_{q},c_h$, a petal face $v_1,b_1,\dots ,b_l=c_1,\dots ,c_h$, and a triangular face $v_1,v_2,b_1$. The edges removed in this step are big face edges, denoted by $E_B$, and the added edges are triangulation edges.

In order to make $[F,T']$ biconnected, note that $[F,T']$ consists of a biconnected component which contains F, called block-component, and a set $\mathcal{T}_B$ of subtrees of $T'$, called tree components, each sharing a cut-vertex with the block component. We remove the tree components $\mathcal{T}_B$ from $[F,T']$ and obtain an instance $[F,T'' \subseteq T']$, that is actually the block component of $[F,T']$. Since the removal of $\mathcal{T}_B$ does not change the degree of the vertices of F and does not create any bad face, $[F,T'']$ is indeed a biconnected instance satisfying the two conditions of Lemma 3. Thus, by adding further triangulation edges we augment it to an inner-triangulated instance $[F,T^{\varDelta }=T'']$.

Lemma 4

Let e=(b, v) be an edge of $E_P \cup E_B$, where $b \in T$ and $v \in F$. Then, either e is a triangulation edge in $[F,T^{\varDelta }]$ or b belongs to a tree component $T_c$ of $\mathcal{T}_B$ sharing a cut-vertex c with $[F,T'']$. In the latter case, (v, c) is a triangulation edge in $[F,T^{\varDelta }]$.

Lemma 5

Let $T_c \in \mathcal{T}_B$ be a tree component such that there exists at least an edge $(b,v) \in E_P \cup E_B$, with $b \in T_c$ and $v \in F$. Then, for each edge in $E_P \cup E_B$ with an endvertex belonging to $T_c$, the other endvertex is v.

Performing the above operations for every cycle-tree graph [F, T] yields an inner-triangulated 2-outerplanar graph $[G,\mathcal{H}^{\varDelta }]$ as outcome of Part A. We then label $[G,\mathcal{H}^{\varDelta }]$ with the algorithm from Sect. 3.2 and describe next how to extend this labeling to $\mathcal{T}_B$.

Part B: We consider the tree components $T_c \in \mathcal {T}_B$ for each face F of G; let $[F,T^{\varDelta }]$ be the corresponding inner-triangulated cycle-tree graph. We label the vertices of $T_c$ and simultaneously augment $[F,T^{\varDelta }]$ with dummy vertices and edges, so that $[F,T^{\varDelta }]$ remains inner-triangulated (and hence can be embedded, by Lemma 2) and the vertices of $T_c$ can be later placed on the petal points of the points where dummy vertices are placed. The face of $[F,T'']$ to which $T_c$ belongs might have been split into several faces of $[F,T^{\varDelta }]$ by triangulation edges. We assign $T_c$ to any of such faces f that is incident to the root c of $T_c$. Then, we label $T_c$ based on the type of f; we distinguish two cases.

Suppose f is a triangular face (c, v, w) with $v,w \in F$ and $c \in T^\varDelta$; assume $\ell (v)<\ell (w)$. We create a path $P_c$ containing $|T_c|-1$ dummy vertices and append $P_c$ at c. Then, we connect every dummy vertex of $P_c$ with both v and w. If $\ell (c) \le \ell (v)$, then we label the vertices of $P_c$ with $\ell (P_c)=\ell (v)$. If $\ell (c) \ge \ell (w)$, then we label $\ell (P_c)=\ell (w)$.

Fig. 4.

(a)–(c) Inserting dummy vertices for a tree-component in a face (a, b, v) with $v \in F$ and $a,b \in T^\varDelta$, when (a) $\ell (a)=\ell (b)$, (b) $\ell (a) \ne \ell (b)$ and $\ell (w) < \ell (v)$, and (c) $\ell (a) \ne \ell (b)$ and $\ell (w) > \ell (v)$. (d)–(e) Moving dummy vertices to petal points if $\ell (a) = \ell (b)$ and if $\ell (a) \ne \ell (b)$ (Color figure online).

Suppose f is a triangular face (a, b, v) with $v \in F$ and $a,b \in T^\varDelta$, refer to Fig. 4; assume $\ell (a) \le \ell (b)$. Replace edge (a, b) with a path $P_c$ between a and b with $|T_c|-1$ internal dummy vertices, and connect each of them to v and to w, where w is the other vertex of F adjacent to both a and b. For each dummy vertex x of $P_c$, we assign $\ell (x) = \ell (a)$ if $\ell (v) \le \ell (a)$; we assign $\ell (x) = \ell (b)$ if $\ell (v) \ge \ell (b)$; and we assign $\ell (x) = \ell (v)$ if $\ell (a) < \ell (v) < \ell (b)$. The existence of edge $(a,b) \in T^\varDelta$ implies that either a is the parent of b in $T^\varDelta$ or vice versa. Suppose the former, the other case is analogous. Then, v and w are the extremal neighbors of b in F, and thus either $\ell (v) \le \ell (b) \le \ell (w)$ or $\ell (w) \le \ell (b) \le \ell (v)$. Also, if $\ell (a) \ne \ell (b)$, then $\ell (a)$ does not lie strictly between $\ell (v)$ and $\ell (w)$. In fact, this can only happen if $\ell (b)$ strictly lies between $\ell (v)$ and $\ell (w)$, and $\ell (a)=\ell (b)$ (which happens only if a is a non-fork vertex). Since $\ell (a) \le \ell (b)$, by assumption, this implies that $\ell (a) \le \ell (v),\ell (w)$. The two observations before can be combined to conclude that, if $\ell (a)=\ell (b)$, then all the tree components lying inside faces (a, b, v) and (a, b, w) have the same label as a and b (Fig. 4(a)). Otherwise, either the tree components inside (a, b, v) have label $\ell (b)$ and those inside (a, b, w) have label $\ell (w)$ (Fig. 4(b)), or the tree components inside (a, b, v) have label $\ell (v)$ and those inside (a, b, w) have label $\ell (b)$ (Fig. 4(c)). All added edges are again triangulation edges.

We apply Part B to every cycle-tree graph of $[G,\mathcal{H}^\varDelta ]$, hence creating an inner-triangulated 2-outerplanar graph $[G,\mathcal{H}^\mathcal{A}]$ where $\mathcal{H}^\mathcal{A}$ is a forest. Since all the dummy vertices of $P_c$ are connected to two vertices $v,w \in F$, they become non-fork vertices. Note that the labeling of the dummy vertices coincides with the one obtained by the algorithm in Sect. 3.2, except for the case when f is a triangular face (a, b, v) with $v \in F$ and $a,b \in T^\varDelta$, and $\ell (a) < \ell (v) < \ell (b)$. In this case the algorithm would have labeled either $\ell (P_c)=\ell (a)$ or $\ell (P_c)=\ell (b)$, depending on whether b is the parent of a or vice versa. However, since $\ell (a) < \ell (v) < \ell (b)$ holds in $[F,T^\varDelta ]$, and since (a, b, v) is a triangular face of $[F,T^\varDelta ]$, no vertex of $[F,T^\varDelta ]$ different from v has the same label as v. Hence, graph $H_i$, for each i, is a tree with at most one vertex of degree 3. We thus apply Lemma 2 to obtain a planar embedding $\varGamma ^\mathcal{A}$ of $[G,\mathcal{H}^\mathcal{A}]$ on S.

4.3 Transformation of the Embedding

We remove the all the triangulation edges added in the construction, and then restore each tree component $T_c$, which is represented by path $P_c$. Since the vertices of $P_c$ are non-fork vertices and have the same label i, by construction, they are placed on the same segment $s \in \{s^+, s^N, s^-\}$ of $S_j$, where $p_j$ is the point vertex $v_i$ is placed on.

We remove all the internal edges of $P_c$ and move each vertex x of $P_c$ from the point p of s it lies on to one of the corresponding petal points, either l(p) or r(p), as follows. Let v be a vertex of G connected to a vertex of $T_c$ by an edge in $E_P \cup E_B$, if any; recall that, by Lemma 5, all the edges of $E_P \cup E_B$ connecting $T_c$ to G are incident to v. If $\ell (x) < \ell (v)$, then move x to r(p); tree components connected to w in Fig. 4(d) and (e). If $\ell (x) > \ell (v)$, then move x to l(p); tree component connected to v in Fig. 4(e). Otherwise, $\ell (x) = \ell (v)$; in this case $s \ne s^N$, by construction, and hence we have to distinguish the following two cases: If $s = s^+$, then move x to l(p), otherwise move x to r(p) (tree components attached to a and b, respectively, and connected to v in Fig. 4(e)). If no vertex $v \in G$ is connected to $T_c$, then move x to r(p) if $\ell (c) < \ell (x)$ (tree component attached to a in Fig. 4(e)), and to l(p) otherwise.

We prove that this operations maintain planarity. The internal edges of $T_c$ do not cross since the petal points, together with the point where c lies, form a convex point set, on which it is possible to construct a planar embedding of every tree [4]. As for the edges connecting vertices of $T_c$ to v, by Lemma 4, v has visibility to the root c of $T_c$, since (v, c) is a triangulation edge; by Property 1, this visibility from v extends to all the segment s where $P_c$ had been placed on; and by the construction of $S^*$, to all the corresponding petal points. The proof for the edges (a, b) that had been subdivided when merging tree component $T_c$ (green edges in Fig. 4(d) and (e)) is in [2].

Claim 1

Reinserting every edge (a, b) such that there existed a path $P_c$ between a and b does not introduce any crossing.

To complete the transformation it remains to insert the edges of $E_P \cup E_B$ which were not inserted in the previous step. Since by Lemma 4 all of these edges were also triangulation edges, their insertion does not produce any crossing.

Lemma 6

There exists a universal point set of size $O(n^{3/2})$ for the class of n-vertex 2-outerplanar graphs $[G,\mathcal{H}]$ where $\mathcal{H}$ is a forest.

5 General 2-Outerplanar Graphs

In this section we give a high-level idea of how to extend the result of Lemma 6 to any arbitrary 2-outerplanar graph $[G,\mathcal{H}]$. The complete description can be found in [2].

The idea is to convert every graph $G_h \in \mathcal{H}$ into a tree $T_h$; embed the resulting graph on $S^*$; and finally revert the conversion from each $T_h$ to $G_h$. Each tree $T_h$ is created by substituting each biconnected block B of $G_h$ by a star, centered at a dummy vertex and with a leaf for each vertex of B, where leaves shared by more stars are identified. This results in a 2-outerplanar graph whose inner level is a forest.

The embedding of this graph on $S^*$ is performed similarly as in Lemma 6, with some slight modifications to the labeling algorithm, especially for the vertices of $T_h$ corresponding to cut-vertices of $G_h$, and to the procedure for merging the tree components. These modifications allow us to ensure that the vertices of each block of $G_h$ lie on a convex portion of $S^*$, where they can thus be drawn without crossings [5, 10].

We finally reduce the size of $S^*$ to $O(n \log n)$ by using the super-pattern sequence $\xi$ from [3], which is a sequence of integers $\xi _j$, with $\sum _{j=1,\dots ,n} \xi _j = O(n \log n)$. Sequence $\xi$ majorizes every sequence of integers that sum up to n. We hence assign the size of each point set $S_j$ based on this sequence, instead of using dense or sparse point sets.

Theorem 1

There exists a universal point set of size $O(n \log n)$ for the class of n-vertex 2-outerplanar graphs.

6 Conclusions

We provided a universal point set of size $O(n \log n )$ for 2-outerplanar graphs. A natural question is whether our techniques can be extended to other meaningful classes of planar graphs, such as 3-outerplanar graphs. We also find interesting the question about the required area of universal point sets. In fact, while the integer grid is a universal point set for planar graphs with $O(n^2)$ points and $O(n^2)$ area, all known point sets of smaller size, even for subclasses of planar graphs, require a larger area. We thus ask whether universal point sets of subquadratic size require polynomial or exponential area.