On the Upward Book Thickness Problem: Combinatorial and Complexity Results

A long-standing conjecture by Heath, Pemmaraju, and Trenk states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are $at$-outerplanar graphs. On the complexity side, it is known that deciding whether a graph has upward book thickness $k$ is NP-hard for any fixed $k \ge 3$. We show that the problem, for any $k \ge 5$, remains NP-hard for graphs whose domination number is $O(k)$, but it is FPT in the vertex cover number.


Introduction
A k-page book embedding (or k-stack layout) of an n-vertex graph G = (V, E) is a pair π, σ consisting of a bijection π : V → {1, . . ., n}, defining a total order on V , and a page assignment σ : E → {1, . . ., k}, partitioning E into k subsets E i = {e ∈ E | σ(e) = i} (i = 1, . . ., k) called pages (or stacks) such that no two edges uv, wx ∈ E mapped to the same page σ(uv) = σ(wx) cross in the following sense.Assume, w.l.o.g., π(u) < π(v) and π(w) < π(x) as well as π(u) < π(w).Then uv and wx cross if π(u) < π(w) < π(v) < π(x), i.e., their endpoints interleave.The book thickness (or stack number) of G is the smallest k for which G admits a k-page book embedding.Book embeddings and book thickness of graphs are well-studied topics in graph drawing and graph theory [11,17,24,29].For instance, it is NP-complete to decide for k ≥ 2 if the book thickness of a graph is at most k [7,11] and it is known that planar graphs have book thickness at most 4 [30]; this bound has recently been shown tight [6].More in general, the book thickness of graphs of genus g is O( √ g) [25] and constant upper bounds are known for some families of non-planar graphs [4,5,18].

arXiv:2108.12327v1 [cs.DM] 27 Aug 2021
Upward book embeddings (UBEs) are a natural extension of book embeddings to directed acyclic graphs (DAGs) with the additional requirement that the vertex order π respects the directions of all edges, i.e., π(u) < π(v) for each uv ∈ E (and hence G must be acyclic).Thus the ordering induced by π is a topological ordering of V .Book embeddings with different constraints on the vertex ordering have also been studied in [2,3,20].Analogously to book embeddings, the upward book thickness (UBT) of a DAG G is defined as the smallest k for which G admits a k-page UBE.The notion of upward book embeddings is similar to upward planar drawings [14,16], i.e., crossing-free drawings, where additionally each directed edge uv must be a y-monotone curve from u to v. Upward book embeddings have been introduced by Heath et al. [22,23].They showed that graphs with UBT 1 can be recognized in linear time, whereas Binucci et al. [9] proved that deciding the UBT of a graph is generally NP-complete, even for fixed values of k ≥ 3. On the positive side, deciding if a graph admits a 2-page UBE can be solved in polynomial time for st-graphs of bounded treewidth [9].
Constant upper bounds on the UBT are known for some graph classes: directed trees have UBT 1 [23], unicyclic DAGs, series-parallel DAGs, and N-free upward planar DAGs have UBT 2 [1,15,23,26].Frati et al. [19] studied UBEs of upward planar triangulations and gave several conditions under which they have constant UBT.Interestingly, upward planarity is a necessary condition to obtain constant UBT, as there is a family of planar, but non-upward planar, DAGs that require Ω(n) pages in any UBE [21].Back in 1999, Heath et al. [23] conjectured that the UBT of outerplanar graphs is bounded by a constant, regardless of their upward planarity.Another long-standing open problem [28] is whether upward planar DAGs have constant UBT; in this respect, examples with a lower bound of 5 pages are known [27] and there is no known upper bound better than O(n).
Contributions.In this paper, we contribute to the research on the upward book thickness problem from two different directions.We first report some notable progress towards the conjecture of Heath et al. [23].We consider subfamilies of upward outerplanar graphs (see Section 2 for definitions), namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are st-outerplanar graphs, and provide constant upper bounds on their UBT (Section 3).Our proofs are constructive and give rise to polynomial-time book embedding algorithms.We then investigate the complexity of the problem (Section 4) and show that for any k ≥ 5 it remains NPcomplete for graphs whose domination number is in O(k).On the positive side, we prove that the upward book thickness problem is fixed-parameter tractable in the vertex cover number.These two results narrow the gap between tractable and intractable parameterizations of the problem.Proofs of statements marked with a ( ) have been sketched or omitted and are in the appendix.

Preliminaries
We assume familiarity with basic concepts in graph drawing (see also [13]).
BC-tree.The BC-tree of a connected graph G is the incidence graph between the (maximal) biconnected components of G, called blocks, and the cut-vertices of G.A block is trivial if it consists of a single edge, otherwise it is non-trivial.
Outerplanarity.An outerplanar graph is a graph that admits an outerplanar drawing, i.e., a planar drawing in which all vertices are on the outer face, which defines an outerplanar embedding.Unless otherwise specified, we will assume our graphs to have planar or outerplanar embeddings.An outerplanar graph G is internally triangulated if it is biconnected and all its inner faces are cycles of length 3.An edge of G is outer if it belongs to the outer face of G, and it is inner otherwise.A cactus is a connected outerplanar graph in which any two simple cycles have at most one vertex in common.Therefore, the blocks of a cactus graph are either single edges (and hence trivial) or cycles.The weak dual G of a planar graph G is the graph having a node for each inner face of G, and an edge between two nodes if and only if the two corresponding faces share an edge.For an outerplanar graph G, its weak dual G is a tree.If G is a path, then G is an outerpath.A fan is an internally-triangulated outerpath whose inner edges all share an end-vertex.
Directed graphs.A directed graph G = (V, E), or digraph, is a graph whose edges have an orientation.We assume each edge e = uv of G to be oriented from u to v, and hence denote u and v as the tail and head of e, respectively.A vertex u of G is a source (resp.a sink) if it is the tail (resp.the head) of all its incident edges.If u is neither a source nor a sink of G, then it is internal.A DAG is a digraph that contains no directed cycle.An st-DAG is a DAG with a single source s and a single sink t; if needed, we may use different letters to denote s and t.A digraph is upward (outer)planar, if it has a (outer)planar drawing such that each edge is a y-monotone curve.Such a drawing (if any) defines an upward (outer)planar embedding.An upward planar digraph G (with an upward planar embedding) is always a DAG and it is bimodal, that is, the sets of incoming and outgoing edges at each vertex v of G are contiguous around v (see also [13]).The underlying graph of a digraph is the graph obtained by disregarding the edge orientations.An st-outerplanar graph (resp.st-outerpath) is an st-DAG whose underlying graph is outerplanar graph (resp.an outerpath).An st-fan is an st-DAG whose underlying graph is a fan and whose inner edges have s as an end-vertex.

Lemma 1 ( ).
Let G be an upward outerplanar graph and let c be a cut-vertex of G. Then there are at most two blocks of G for which c is internal.Basic operations.Let π and π be two orderings over vertex sets V and V ⊆ V , respectively.Then π extends π if for any two vertices u, v ∈ V with π (u) < π (v), it holds π(u) < π(v).We may denote an ordering π as a list v 1 , v 2 , . . ., v |V | and use the concatenation operator • to define an ordering from other lists, e.g., we may obtain the ordering π and π 2 = v 3 , v 4 .Also, let π be an ordering over V and let u ∈ V .We denote by π u − and π u + the two orderings such that π = π u − • u • π u + .Consider two orderings π over V and π over V with V ∩ V = {u, v}, and such that: (i) u and v are consecutive in π, and (ii) u and v are the first and the last vertex of π , respectively.The ordering π * over V ∪ V obtained by merging π and π is π Note that π * extends both π and π .

Book thickness of outerplanar graphs
In this section, we study the UBT of three families of upward outerplanar graphs.We begin with internally-triangulated upward outerpaths (Section 3.1), which are biconnected and may have multiple sources and sinks.We then continue with families of outerplanar graphs that are not biconnected but whose biconnected components have a simple structure, namely outerplanar graphs whose biconnected components are st-DAGs (Section 3.2), and cactus graphs (Section 3.3).

Internally-triangulated upward outerpaths
In this subsection, we assume our graphs to be internally triangulated.We will exploit the following definition and lemmas for our constructions.Definition 1.Let G be an st-outerpath and uv be an outer edge different from st.An UBE π, σ of G is uv-consecutive if the following properties hold: (i) u and v are consecutive in π, (ii) the edges incident to s lie on one page, and (iii) the edges incident to t lie on at most two pages.
An st-outerplanar graph is one-sided if the edge st is an outer edge.

Lemma 2.
Let G be a one-sided st-outerplanar graph.Then, G admits a 1-page UBE π, σ that is uv-consecutive for each outer edge uv = st.
Proof.Consider the path of the outer face of G that encompasses all vertices of G and does not contain the edge st.Let π, σ be the 1-page UBE in which π is the ordering defined by such path.One easily verifies that no two edges cross and that the endpoints u, v of each outer edge uv = st are consecutive in π.
Lemma 3. Let G be an st-fan and let uv = st be an outer edge of G.Then, G admits a 2-page uv-consecutive UBE.
Proof.Let P = {s, a 1 , . . ., a , t} and P r = {s, b 1 , . . ., b r , t} be the left and right st-paths of the outer face of G, respectively.Then the edge uv belongs to either P or P r .We show how to construct an UBE π, σ of G that satisfies the requirements of the lemma, when uv belongs to P (see Fig. 1); the construction when uv belongs to P r is symmetric (it suffices to flip the embedding along st).Since uv = st, we have either (a) u = s and v = t, or (b) v = t and u = s, or (c) {u, v} ∩ {s, t} = ∅.In case (a), refer to Fig. 1(a).We set π = s, a 1 , . . ., a , b 1 , . . ., b r , t (that is, we place P before P r ), σ(e) = 1 for each edge e = a t, and σ(a t) = 2.In case (b), refer to Fig. 1(b).We set π = s, b 1 , . . ., b r , a 1 , . . ., a , t (that is, we place P after P r ), σ(e) = 1 for each edge e = b r t, and σ(b r t) = 2.In case (c), we can set π and σ as in any of case (a) and (b).In all three cases, all outer edges (including uv) are consecutive, except for one edge e incident to t; also, all edges (including those incident to s) are assigned to the same page, except for e, which is assigned to a second page.
The next definition allows us to split an st-outerpath into two simpler graphs.
The extreme faces of an st-outerpath G are the two faces that correspond to the vertices of G having degree one.
Definition 2. An st-outerpath G is primary if and only if the path forming G has one extreme face incident to s.
Let G be an st-outerpath and refer to Fig. 2(a).Consider the subgraph F s of G induced by s and its neighbors, note that this is an sw-fan.Assuming F s = G , and since G is an outerpath, one or two edges on the outer face of F s are separation pairs for G .In the former case, it follows that G is primary, since G is a path having one face of F s as its extreme face.In the latter case, since G has a single source s and a single sink t, (at least) one of the two separation pairs splits G into a one-sided uv-outerpath H 1 (for some vertices u, v of F s ) and into a primary st-outerpath G.We will call H 1 , the appendage at uv of G .
Let G be an st-outerpath (not necessarily primary).Consider a subgraph F of G that is an xy-fan (for some vertices x, y of G).Let f 1 , . . ., f h be the ordered list of faces forming the path G.Note that F is the subgraph of G formed by a subset of faces that are consecutive in the path f 1 , . . ., f h .Let f i be the face of F with the highest index.We say that F is incrementally maximal if i = h or F ∪ f i+1 is not an xy-fan.We state another key definition; refer to Fig. 2(b).Definition 3.An st-fan decomposition of an st-outerpath G is a sequence of s i t i -fans F i ⊆ G, with i = 1, . . ., k, such that: (i) F i is incrementally maximal; (ii) For any 1 ≤ i < j ≤ k, F i and F j do not share any edge if j > i + 1, while F i and F i+1 share a single edge, which we denote by e i ; (iii) s 1 = s; (iv) the tail of e i is s i+1 ; (v) edge e i = s i t i ; and (vi) We next show that primary st-outerpaths always have st-fan decompositions.
Proof (Sketch).By Definition 2, one extreme face of G is incident to s; we denote such face by f 1 , and the other extreme face of G by f h .We construct the st-fan decomposition as follows.We initialize F 1 = f 1 and we parse the faces of G in the order defined by G from f 1 to f h .Let F i be the current s i t i -fan and let f j be the last visited face (for some 1 < j < h).
Lemma 5 ( ).Let G be a primary st-outerpath and let F 1 , F 2 , . . ., F k be an st-fan decomposition of G. Any two fans F i and F i+1 are such that if F i+1 is not one-sided, then e i = s i+1 t i .Lemma 6.Let G be a primary st-outerpath and let F 1 , F 2 , . . ., F k be an st-fan decomposition of G. Also, let e = s k t k be an outer edge of F k .Then, G admits a 4-page e-consecutive UBE π, σ .
Proof.We construct a 4-page e-consecutive UBE of G by induction on k; see also Fig. 2(c), which shows a UBE of the primary st-outerpath in Fig. 2(b).
Suppose k = 1.Then G consists of the single s 1 t 1 -fan and a 2-page econsecutive UBE of G exists by Lemma 3.
Suppose now k > 1.Let G i be the subgraph of G induced by Recall that e i is the edge shared by F i and F i+1 and that the tail of e i coincides with s i+1 .Let π, σ be a 4-page e k−1 -consecutive UBE of G k−1 , which exists by induction since e k−1 = s k−1 t k−1 by condition (v) of Definition 3, and distinguish whether F k is one-sided or not.
If F k is one-sided, it admits a 1-page e-consecutive UBE π , σ by Lemma 2. In particular, e = e k−1 , since e k−1 is not an outer edge of G k .Also, e k−1 = s k t k , and hence the two vertices shared by π and π are s k , t k , which are consecutive in π and are the first and the last vertex of π .Then we define a 4-page e-consecutive UBE π * , σ * of G k as follows.The ordering π * is obtained by merging π and π.Since e k−1 = s k t k is uncrossed (over all pages of σ), for every edge e of F k , we set σ * (e) = σ(e k−1 ), while for every other edge e we set σ * (e) = σ(e).
If F k is not one-sided, by Lemma 3, F k admits a 2-page e-consecutive UBE π , σ .Then we obtain a 4-page e-consecutive UBE π * , σ * of G k as follows.By Lemma 5, it holds e k−1 = s k t k−1 , and thus s k and t k−1 are the second-to-last and the last vertex in π, respectively; also, s k is the first vertex in π .We set by Definition 1 we know that the edges incident to t k−1 can use up to two different pages.On the other hand, these are the only edges that can be crossed by an edge of F k assigned to one of these two pages.Therefore, in Lemma 3, we can assume σ uses the two pages not used by the edges incident to t k−1 , and set σ * (e) = σ (e) for every edge e of F k and σ * (e) = σ(e) for every other edge.
Next we show how to reinsert the appendage H 1 (Lemma 8).To this aim, we first provide a more general tool (Lemma 7) that will be useful also in Section 3.2.
Lemma 7 ( ).Let G = (V, E) be a primary st-outerpath with a 4-page econsecutive UBE of G obtained by using Lemma 6, for some outer edge e of G.
We now define a decomposition of an upward outerpath G; refer to Fig. 3. Let P ⊂ G be an st-outerpath, then P is the subgraph of G formed by a subset of consecutive faces of G = f 1 , . . ., f h .Let f j and f j be the faces of P with the smallest and highest index, respectively.Let F i be the incrementally maximal e 1 Fig. 3.An st-outerpath decomposition of an upward outerpath; edges ei are fat (e1 s i t i -fan of P i (assuming f j , . . ., f j to be the ordered list of faces of P i ).We say that P is incrementally maximal if i = h or if P ∪ f i+1 is not an st-outerpath or if P ∪ f i+1 is still an st-outerpath but the edge s i t i of F i is an outer edge of P .Definition 4.An st-outerpath decomposition of an upward outerpath G is a sequence of s i t i -outerpaths P i ⊆ G, with i = 1, 2, . . ., m, such that: (i) P i is incrementally maximal; (ii) For any 1 ≤ i < j ≤ m, P i and P j share a single edge if j = i + 1, which we denote by e i , while they do not share any edge otherwise; and (iii) Lemma 9 ( ).Every upward outerpath admits an st-outerpath decomposition.
An st-outerpath that is not a single st-fan is called proper in the following.Let P 1 , P 2 , . . ., P m be an st-outerpath decomposition of an upward outerpath G, two proper outerpaths P i and P j are consecutive, if there is no proper outerpath P a , such that i < a < j.We will use the following technical lemmas.
Lemma 10 ( ).Two consecutive proper outerpaths P i and P j share either a single vertex v or the edge e i .In the former case, it holds j > i + 1, in the latter case it holds j = i + 1.
Lemma 11 ( ).Let P i and P j be two consecutive proper outerpaths that share a single vertex v. Then each P a , with i < a < j, is an s a t a -fan such that v = s a if v is the tail of e i , and v = t a otherwise.Lemma 12 ( ).Let v be a vertex shared by a set P of n P outerpaths.Then at most two outerpaths of P are such that v is internal for them, and P contains at most four proper outerpaths.We are now ready to prove the main result of this section.
Proof (Sketch).Let P 1 , P 2 , . . ., P m be an st-outerpath decomposition of G (Lemma 9).Based on Lemma 10, a bundle is a maximal set of outerpaths that either share an edge or a single vertex.Let G b be the graph induced by the first b bundles of G (going from P 1 to P m ).We prove the statement by induction on the number l of bundles of G.In particular, we can prove that G l admits a e g(l)consecutive 16-page UBE π l σ l , where g(l) is the greatest index such that P g(l) belongs to G l , and such that each single s i t i -outerpath uses at most 4 pages.In the inductive case, we distinguish whether the considered bundle contains only two outerpaths that share an edge, or at least three outerpaths that share a vertex.Here, we exploit the crucial properties of Lemmas 8, 11 and 12, which allow us to limit the interaction between different outerpaths in terms of pages.

Upward outerplanar graphs
We now deal with upward outerplanar graphs that may be non-triangulated and may have multiple sources and sinks, but whose blocks are st-DAGs.We begin with the following lemma, which generalizes Lemma 8 in terms of UBT.
Proof (Sketch).By exploiting a technique in [14], we can assume that G is internally triangulated.Let f 1 , . . ., f h be a path in G whose primal graph P ⊂ G is a primary st-outerpath.Each outer edge uv of P is shared by P and by a one-sided uv-outerpath.Then a 4-page UBE of G exists by Lemma 7.
We are now ready to show the main result of this subsection.
Theorem 2 ( ).Every upward outerplanar graph G whose biconnected components are st-outerplanar graphs admits an 8-page UBE.Proof (Sketch).We prove a stronger statement.Let T be a BC-tree of G rooted at an arbitrary block ρ, then G admits an 8-page UBE π, σ that has the pageseparation property: For any block β of T , the edges of β are assigned to at most 4 different pages.We proceed by induction on the number h of cut-vertices in G.If h = 0, then G consists of a single block and the statement follows by Lemma 13.Otherwise, let c be a cut-vertex whose children are all leaves.Let ν 1 , . . ., ν m be the m > 1 blocks representing the children of c in T , and let µ be the parent block of c.Also, let G be the maximal subgraph of G that contains µ but does not contain any vertex of ν 1 , . . ., ν m except c, that is, By induction, G admits an 8-page UBE π , σ for which the page-separation property holds, as it contains at most h − 1 cut-vertices.On the other hand, each ν i admits a 4-page UBE π i , σ i by Lemma 13.By Lemma 1, at most two blocks in {µ} ∪ {ν 1 , . . ., ν m } are such that c is internal.
Let us assume that there are exactly two such blocks and one of these two blocks is µ, as otherwise the proof is just simpler.Also, let ν a , for some 1 ≤ a ≤ m be the other block for which c is internal.Up to a renaming, we can assume that ν 1 , . . ., ν a−1 are st-outerplanar graphs with sink c, while ν a+1 , . . ., ν m are stouterplanar graphs with source c.Refer to Fig. 4. Crucially, we set: The page assignment is based on the fact that e and e , such that e ∈ ν i and e ∈ G , cross each other only if i = a and in such a case e is incident to c.

Upward cactus graphs
The first lemma allows us to consider cactus graphs with no trivial blocks.

Lemma 14 ( ).
A cactus G can always be augmented to a cactus G with no trivial blocks and such that the embedding of G is maintained.
It is well known that any DAG whose underlying graph is a cycle admits a 2-page UBE [23,Lemma 2.2].We can show a slightly stronger result, which will prove useful afterward; see Fig. 5(b).
Lemma 15 ( ).Let G be a DAG whose underlying graph is a cycle and let s be a source (resp.let t be a sink) of G.Then, G admits a 2-page UBE π, σ where s is the first vertex (resp.t is the last vertex) in π.
Using the proof strategy of Theorem 2, we can exploit Lemma 15 to show: Theorem 3 ( ).Every upward outerplanar cactus G admits a 6-page UBE.
Proof (Sketch).By Lemma 14, we can assume that all the blocks of G are nontrivial, i.e., correspond to cycles.Also, let T be the BC-tree of G rooted at any block.The theorem can be proved by induction on the number of blocks in T .In fact, we prove the following slightly stronger statement: G admits a 6-page UBE in which the edges of each block lie on at most two pages.The proof crucially relies on Lemma 1 and follows the lines of Theorem 2.

Complexity Results
Recall that the upward book thickness problem is NP-hard for any fixed k ≥ 3 [9].This implies that the problem is para-NP-hard, and thus it belongs neither to the FPT class nor to the XP class, when parameterized by its natural parameter (unless P=NP).In this section, we investigate the parameterized complexity of the problem with respect to the domination number and the vertex cover number, showing a lower and an upper bound, respectively.

Hardness Result for Graphs of Bounded Domination Number
A domination set for a graph G = (V, E) is a subset D ⊆ V such that every vertex in V \D has at least one neighbor in D. The domination number γ(G) of G is the number of vertices in a smallest dominating set for G.Given a DAG G such that UBT(G) ≤ k, one may consider the trivial reduction obtained by considering the DAG G obtained from G by introducing a new super-source (connected to all the vertices), which has domination number 1 and for which it clearly holds that UBT(G ) ≤ k + 1.However, the other direction of this reduction is not obvious, and indeed for this to work we show a more elaborated construction.Theorem 4 ( ).Let G be an n-vertex DAG and let k be a positive integer.It Proof (Sketch).The proof is based on the construction in Fig. 6.We obtain graph G by suitably combining G with an auxiliary graph H whose vertices have the same order in any UBE of H, and UBT(H) = k + 2. The key property of G is that the vertices of G are incident to vertices c and f of H, and that edges incident to each of these vertices must lie in the same page in any UBE of G .Since testing for the existence of a k-page UBE is NP-hard when k ≥ 3 [9], Theorem 4 implies that the problem remains NP-hard even for inputs whose domination number is linearly bounded by k.We formalize this in the following.
Theorem 5.For any fixed k ≥ 5, deciding whether an st-DAG G is such that Theorem 5 immediately implies that the upward book thickness problem parameterized by the domination number is para-NP-hard.On the positive side, we next show that the problem parameterized by the vertex cover number admits a kernel and hence lies in the FPT class.

FPT Algorithm Parameterized by the Vertex Cover Number
We prove that the upward book thickness problem parameterized by the vertex cover number admits a (super-polynomial) kernel.We build on ideas in [8].A vertex cover of a graph G = (V, E) is a subset C ⊆ V such that each edge in E has at least one incident vertex in C (a vertex cover is in fact a dominating set).The vertex cover number of G, denoted by τ , is the size of a minimum vertex cover of G. Deciding whether an n-vertex graph G admits vertex cover of size τ , and if so computing one, can be done in O(2 τ + τ • n) time [10].Let G = (V, E) be an n-vertex DAG with vertex cover number τ .Let C = {c 1 , c 2 , . . ., c τ } be a vertex cover of G such that |C| = τ .The next lemma matches an analogous result in [8].
Lemma 16 ( ).G admits a τ -page UBE that can be computed in O(τ • n) time.
For a fixed k ∈ N, if k ≥ τ , then G admits a k-page UBE by Lemma 16.Thus we assume k < τ .Two vertices u, v ∈ V \ C are of the same type U if they have the same set of neighbors U ⊆ C and, for every w ∈ U , the edges connecting w to u and w to v have the same orientation.We proceed with the following reduction rule.For each type U , let V U denote the set of vertices of type U .Since there are 2 τ different neighborhoods of size at most τ , and for each of them there are at most 2 τ possible orientations, the type relation yields at most 2 2τ distinct types.Therefore assigning a type to each vertex and applying R.1 exhaustively can be done in 2 O(τ ) + τ • n time.We can prove that the rule is safe.

Open Problems
The next questions naturally arise from our research: (i) Is the UBT of upward outerplanar graphs bounded by a constant?(ii) Are there other parameters that are larger than the domination number (and possibly smaller than the vertex cover number) for which the problem is in FPT? (iii) Does the upward book thickness problem parameterized by vertex cover number admit a polynomial kernel?
A Missing Proofs of Section 2 Lemma 1 ( ).Let G be an upward outerplanar graph and let c be a cut-vertex of G. Then there are at most two blocks of G for which c is internal.
Proof.Observe that if c is internal for a block β of G, then β must be non-trivial.Suppose, for a contradiction, that there exist three non-trivial blocks β 1 , β 2 , and β 3 of G incident to c for which c is internal.Let E be the upward outerplanar embedding of G. Since each β i is non-trivial, it contains a cycle of edges C i , for which c is internal.Let e i and h i be the edges of C i for which c is the head and the tail, respectively.Assume, w.l.o.g., that e 1 , e 2 , and e 3 appear in this left-to-right order in E.Then, by the upward planarity of E, it holds that the edges h 1 , h 2 , and h 3 appear in this left-to-right order in E. Thus, by planarity, C 2 must enclose in its interior all the vertices of either C 1 or C 3 (except for c), contradicting the fact that E is outerplanar.

B Missing Proofs of Section 3
B.1 Missing Proofs of Section 3.1 Lemma 4 ( ).Every primary st-outerpath G admits an st-fan decomposition.
Proof.By Definition 2, one extreme face of G is incident to s; we denote such face by f 1 , and the other extreme face of G by f h .We construct the st-fan decomposition as follows.We initialize F 1 = f 1 and we parse the faces of G in the order defined by G from f 1 to f h .Let F i be the current s i t i -fan and let f j be the last visited face (for some 1 < j < h).If F i ∪ f j+1 is an s i t i -fan, we set F i = F i ∪ f j+1 , otherwise we finalize F i and set F i+1 = f j .
We now prove that the computed set of fans, denoted by F 1 , F 2 , . . ., F k , is indeed an st-fan decomposition of G.If k = 1, one easily verifies that all conditions of Definition 3 are satisfied.Suppose now that k > 1.To prove condition (i), observe that for a fan F i to be not incrementally maximal, F i+1 (if i < k) must contain a face f such that F i ∪ f is an s i t i -fan.However, F i+1 cannot contain f by construction, since we iteratively expanded F i towards f h until it was no further possible.Condition (ii) is satisfied because we parse the faces of G one by one and never assign the same face to two fans.Condition (iii) is satisfied because the first face f 1 considered is incident to s. Condition (iv) is satisfied because F i and F i+1 share edge e i and if the tail of e i were not s i+1 , then G would contain another source distinct from s. Concerning condition (v), let f be the face of F i+1 incident to e i .If e i = s i t i , by (iv) it holds s i = s i+1 , and therefore F i ∪ f would still be an s i t i -fan, which is not possible by construction.Condition (vi) is satisfied unless there exists some face f that, in G, comes before f 1 and hence is not parsed by our construction.However this is not possible because f 1 and f h are the two extreme faces of G.

Lemma 5 ( ).
Let G be a primary st-outerpath and let F 1 , F 2 , . . ., F k be an st-fan decomposition of G. Any two fans F i and F i+1 are such that if F i+1 is not one-sided, then e i = s i+1 t i .u i v i also crosses u j v j and σ * (e) = σ * (e ) because σ * (e) = σ * (u i v i ), σ * (e ) = σ * (u j v j ), and σ * (u i v i ) = σ * (u j v j ).Lemma 8 ( ).Let G = G ∪ H 1 be an st-outerpath, such that G is a primary st-outerpath and H 1 is the appendage at uv of G .Let F 1 , F 2 , . . ., F k be an st-fan decomposition of G. Let e be an outer edge of G that belongs to either F k or H 1 , or to F 1 if H 1 = ∅.Also, if e ∈ F k , then e = s k t k , otherwise e = s 1 t 1 .Then, G admits an e-consecutive 4-page UBE π, σ .
Proof.If e is an outer edge of F k , since the appendage H 1 of G is a one-sided uv-outerpath that only shares the edge uv with G, the lemma is an immediate consequence of Lemma 7. If e is an outer edge of F 1 or H 1 , we first reverse the orientation of the edges of G and obtain a ts-outerpath G .In a ts-fan decomposition of the primary ts-outerpath of G , which we denote by F 1 , F 2 , . . ., F k , the edge e now belongs to F k (because it belongs to H 1 or to Thus again we can compute a 4-page e-consecutive UBE π , σ .By reversing the ordering π , we obtain a 4-page e-consecutive UBE of G .Lemma 9 ( ).Every upward outerpath admits an st-outerpath decomposition.
Proof.Let G be an internally-triangulated upward outerpath.First, consider the weak dual G = f 1 , . . ., f h of G and the outerpaths P 1 , P 2 , . . ., P m constructed as follows.We initialize P 1 = f 1 and we parse the faces of G in the order defined by G from f 1 to f h .Let P i be the current s i t i -outerpath and let f j be the last visited face (for some 1 < j < h).We set P i = P i ∪ f j if P i ∪ f j is still a singlesource single-sink outerpath, otherwise we set P i+1 = f j and proceed.Clearly, at the end of this process, we have obtained a sequence of s i t i -outerpaths that satisfies Properties (ii) and (iii) of Definition 4. Concerning Property (i), while any of the obtained outerpaths cannot be expanded with the first fan of the next outerpath, it may not hold that e i = s i t i .Thus, we next show how to (re)assign some fans to a different outerpath in P 1 , . . ., P m so to also satisfy Property (i) of Definition 4.
For i = 1, . . ., m − 1, consider a partition of each P i into incrementally maximal st-fans, obtained by visiting the faces of P i from the face f j that contains e i−1 to the face f j that contains e i .Recall that we denoted by F i the last incrementally maximal s i t i -fan of P i and hence e i belongs to F i ; furthermore, let us denote by Q i the incrementally maximal fan, if any, preceding F i in the above partition.Suppose that Property (i) is not satisfied by P 1 , P 2 , . . ., P m , and let j be the minimum index such that e j = s j t j .Consider the two faces of G incident to e j .Observe that one of these faces, which we denote by f , belongs to F j (of P j ), while the other face, which we denote by f , belongs to the first incrementally maximal fan of P j+1 .First, observe that P j = F j , since F j ∪ f is a single-source single-sink outerpath (and therefore the edge s i t i would not be the shared edge of two distinct outerpaths in P 1 , P 2 , . . ., P m ).Therefore, P j also contains Q j .The edge e shared by Q j and F j is not s j t j , as if this were the case, by extending Q j with the face of F j incident to e we would obtain a larger single-source single-sink fan, which contradicts the fact that Q j is incrementally maximal.We update the decomposition P 1 , P 2 , . . ., P m as follows.We remove from P j all the vertices and edges of F j , except for e and its end-vertices.We then set P j+1 = F j ∪ P j+1 , that is, we move the fan F j from P j to P j+1 (note that F j may merge with the first incrementally maximal fan of P j+1 ).See, for instance, the blue-orange pair and the pink-cyan pair of outerpaths in Fig. 7, and how they are updated in Fig. 3. Observe that e = s j t j is now the unique edge shared by P j and P j+1 .Therefore, s j t j = e j is now satisfied by the pair P j , P j+1 .Repeating the above update, as long as there exists a pair of singlesource single-sink outerpaths in P 1 , P 2 , . . ., P m sharing an edge and violating Property (i), eventually yields the desired st-outerpath decomposition.
Lemma 10 ( ).Two consecutive proper outerpaths P i and P j share either a single vertex v or the edge e i .In the former case, it holds j > i + 1, in the latter case it holds j = i + 1.
Proof.By Definition 4, P i and P j share at most one edge.We first prove that P i and P j cannot be vertex-disjoint.Namely, if j = i + 1, then P i and P j share an edge by Property (ii) of Definition 4. If j > i + 1, let P i+1 , . . ., P j−1 be the non-proper outerpaths between P i and P j .Since each of these non-proper outerpaths are single-source single-sink fans, it must be that they either have different sources or different sinks, and hence they all share the same sink or the same source, respectively, and this common vertex is also common to P i and P j .
Lemma 11 ( ).Let P i and P j be two consecutive proper outerpaths that share a single vertex v. Then each P a , with i < a < j, is an s a t a -fan such that v = s a if v is the tail of e i , and v = t a otherwise.
Proof.We know that each P a , with i < a < j, is an s a t a -fan because P i and P j are consecutive.As already observed in the proof of Lemma 10, since each P a is an s a t a -fan, it must be that all of them either have different sources or different sinks, and hence vertex v is either a common sink or a common source for all of them.Consequently, if v is the tail of e i , then v is the source s a of each P a , otherwise v is the sink t a of each P a .
Lemma 12 ( ).Let v be a vertex shared by a set P of n P outerpaths.Then at most two outerpaths of P are such that v is internal for them, and P contains at most four proper outerpaths.
Proof.Since all outerpaths of P share vertex v, the fact that P contains at most two outerpaths for which v is internal immediately follows by the fact that G has an upward outerplanar embedding.Namely, by bimodality, at most two faces incident to v are such that v is neither the source nor the sink of that face [13].These two faces belong to two different outerpaths P and P in P, and if there existed a third one P with the property of v being internal, then at least one vertex of P or P would not belong to the outer face of G.
To prove the latter part of the statement, observe that, by Lemma 10, it must be that the outerpaths in P form a contiguous subset P i , . . ., P j , with j −i = n P , in the st-outerpath decomposition P 1 , . . ., P m .Since each outerpath in P shares v, in order for G to be a path, it must be that each P a is a fan, except for possibly P i and P j , although not necessarily a fan in which the vertex shared by the inner edges is its source vertex.However, we proved above that at most two of these fans are such that v is internal.Hence, the only elements of P that can be proper are P i , P j and the two for which v is internal.
Proof.Let P 1 , P 2 , . . ., P m be an st-outerpath decomposition of G, which exists by Lemma 9. Based on Lemma 10, a bundle is a maximal set of outerpaths that either share an edge or a single vertex.Let G b be the graph induced by the first b bundles of G (from P 1 to P m ).We prove by induction on the number l of bundles of G that G l admits a e g(l) -consecutive 16-page UBE π l σ l , where g(l) is the largest index such that P g(l) belongs to G l , and such that each single s i t i -outerpath uses at most 4 pages.
Before giving the inductive proof, we observe that each s i t i -outerpath P i , with i = 1, . . ., m, admits an e i -consecutive 4-page UBE π, σ , which can be obtained by using Lemma 8.In particular, let F 1 , . . ., F k be the st-fan decomposition given in input to Lemma 8, and denote by s j , t j the source and sink of F j , respectively.We have three possible cases: (a) F i = F k , or (b) F i = F 1 (and hence P i is primary), or (c) F i belongs to the appendage H 1 of P i .In all the three cases, the preconditions of Lemma 8 applies.Namely, in case (a) s i t i = s k t k and e = s i t i because P i is incrementally maximal; in case (b) s i t i = s 1 t 1 and again e = s i t i because P i is incrementally maximal; in case (c) if must be that s i t i belongs to H 1 (otherwise H 1 would not contain F i ), and hence clearly s i t i = s 1 t 1 .
In the base case l = 0, then m = 1, and an e 1 -consecutive 4-page UBE π 0 σ 0 of P 1 can be obtained by using Lemma 8 as observed before.If l > 0, let B = {P g (l) , . . ., P g(l) } be the l-th bundle of G l .Let G l−1 be the graph obtained by removing from G l the outerpaths in B, except for P g (l) (which belongs to the bundle before B as well, if any).Note that G l−1 contains l − 1 bundles.By induction, let π l−1 , σ l−1 be an e g(l−1) -consecutive 16-page UBE of G l−1 .We distinguish whether B contains only two outerpaths that share an edge, or it contains at least three outerpaths that share a vertex.CASE 1. B contains only two outerpaths P g(l)−1 and P g(l) that share an edge e g(l)−1 = uv.Let π, σ be a e g(l) -consecutive 4-page UBE of P g(l) , obtained by applying Lemma 8, as already observed.Referring to Fig. 8, we set π Clearly, π l extends both π l−1 and π.One easily verifies that if e and e are such that e ∈ G l−1 and e ∈ P g(l) , then e is incident to either u or v. Consequently, e belongs to P g(l)−1 .Namely, if e belonged to another outerpath P = P g(l)−1 , then P, P g(l)−1 , P g(l) would all share a vertex and hence would belong to the same bundle B by the maximality of B. Hence we can set σ l (e) = σ l−1 (e) for each edge e ∈ G l−1 , while, for the edges of P g(l) , we can use any set of 4 pages in σ l−1 not used by the edges of P g(l)−1 .CASE 2. B contains at least three outerpaths that share a vertex.For ease of notation, let us denote P i = P g (l) and P j = P g(l) , and denote by v the shared vertex.By Lemma 12, we know that in P i , . . ., P j there are at most four proper outerpaths, and, by Lemma 11, that between any two consecutive proper outerpaths there are non-proper outerpaths for which v is either a common source or a common sink.We assume that all these four proper outerpaths exist, as the proof is simpler otherwise.For ease of notation, we denote by IN and OUT the first and the last proper outerpaths (in the order from P 1 to P m ).Note that G l−1 contains IN.Also, we denote by EAST and WEST the proper outerpaths such that EAST comes before WEST.Moreover, we denote by IN-EAST fans, EAST-WEST fans, and WEST-OUT fans, the non-proper outerpaths between the IN and EAST, EAST and WEST, WEST and OUT, respectively.For each of them, we compute a UBE by Lemma 8 as in CASE 1, and we merge the obtained UBEs one after the other as in CASE 1.A schematic illustration of the obtained ordering is illustrated in Fig. 9.One can verify that the edges of two fans do not cross unless they belong to the same set (IN-EAST, EAST-WEST, OUT-WEST) and share an edge.Thus, for all non-proper outerpaths we will use two alternating sets of 2 pages each (rather than 4, as these are simple fans), different from Proof.Di Battista and Tamassia [14] proved that every upward planar graph G can be augmented, by only introducing edges, to an upward planar triangulation.In fact, their technique can also be applied to augment an upward outerplanar graph to an internally-triangulated upward outerplanar graph.Thus, we can assume that G is internally triangulated.
Refer to Fig. 10.Let f 1 be any inner face of G that contains vertex s, and let f h be any inner face of G that contains vertex t.Let Π = f 1 , . . ., f h be a path in G.By construction, the primal graph P of Π is a primary st-outerpath.Each edge uv on the outer face of P either belongs to the outer face of G or uv is an inner edge of G and therefore {u, v} is a separation pair of G.We denote s t Fig. 10.Illustration for the proof of Lemma 13.An st-outerplanar graph G; an stouterpath P of G is highlighted with a striped orange background, while the three corresponding subgraphs are light purple, green, and blue.
by H uv the maximal connected component of G that contains uv and no further vertex of P (such component is unique by outerplanarity).Since G has a single source and a single sink, H uv is an outerplanar graph with a single source u and a single sink v.Moreover, since edge uv is on the outer face of H uv , it follows that H uv is a one-sided uv-outerplanar graph.Then a 4-page UBE of G exists by Lemma 7.
Proof.We prove a stronger statement.Let T be a BC-tree of G rooted at an arbitrary block ρ, then G admits an 8-page UBE π, σ that has the page-separation property: For any block β of T , the edges of β are assigned to at most 4 different pages.We proceed by induction on the number h of cut-vertices in G.
If h = 0, then G consists of a single block which is an st-outerplanar graph and the statement follows by Lemma 13.Otherwise, let c be a cut-vertex whose children are all leaves.Let ν 1 , . . ., ν m be the m > 1 blocks representing the children of c in T , and let µ be the parent block of c.Also, let G be the maximal subgraph of G that contains µ but does not contain any vertex of ν 1 , . . ., ν m except c, that is, By induction, G admits an 8-page UBE π , σ for which the page-separation property holds, as it contains at most h − 1 cut-vertices.On the other hand, each ν i admits a 4-page UBE π i , σ i by Lemma 13.By Lemma 1, at most two blocks in {µ} ∪ {ν 1 , . . ., ν m } are such that c is internal.
Let us assume that there are exactly two such blocks and one these two blocks is µ, as otherwise the proof is just simpler.Also, let ν a , for some 1 ≤ a ≤ m be the other block for which c is internal.Up to a renaming, we can assume that ν 1 , . . ., ν a−1 are st-outerplanar graphs with sink c, while ν a+1 , . . ., ν m are st-outerplanar graphs with source c.Refer to Fig. 4. We set: Observe that π extends each π i , as well as π .Then one easily verifies that no two edges e and e such that e ∈ ν i and e ∈ ν j , for any i = j, cross each other in π.
Similarly, two edges e and e such that e ∈ ν i and e ∈ G , cross each other only if i = a and in such a case e is incident to c. Concerning the page assignment, we let σ(e) = σ (e) for each edge e of G .Also, we let σ(e) = σ i (e) for each edge e of ν i , except the edges of ν a , assuming that the set of pages of σ i is the same (up to a renaming) as the one used in σ for the edges of µ (whose size is 4 by the page-separation property).For the edges of ν a , we set σ(e) = σ a (e), assuming that the set of pages of σ a is the same (up to a renaming) as the one in σ not used for the edges of µ (whose size is again four).As we already observed, if two edges e and e such that e ∈ ν a and e ∈ G cross each other, then e is incident to c and hence belongs to µ.Then our assignment guarantees σ(e) = σ(e ).Also, the computed 8-page UBE respects the page-separation property.

B.3 Missing Proofs of Section 3.3 Lemma 14 ( ).
A cactus G can always be augmented to a cactus G with no trivial blocks and such that the embedding of G is maintained.
Proof.If the blocks of G are cycles, then there is nothing to be done.Otherwise, G contains a trivial block β corresponding to the directed edge uv.We show how to extend G to a cactus G , whose number of trivial blocks is one less than the number of trivial blocks of G .For simplicity, we construct G , by showing how to extend an upward outerplanar drawing Γ of G to an upward outerplanar drawing Γ of G .First, we initialize Γ = Γ .Second, we place in Γ a new vertex w above u and below v and arbitrarily close to the drawing of the edge uv.Finally, we draw two new edges uw and wv as y-monotone curves so that uv, uw and wv form a face of the drawing.Clearly, Γ is an upward outerplanar drawing of G , G is a cactus, and G contain one less trivial block of G .By repeating the process until no trivial blocks exist, we eventually obtain G.
Lemma 15 ( ).Let G be a DAG whose underlying graph is a cycle and let s be a source (resp.let t be a sink) of G.Then, G admits a 2-page UBE π, σ where s is the first vertex (resp.t is the last vertex) in π.
Proof.Let sw be any of the two edges incident to s.Let G be the path G − sw.
Since G is a tree, it admits a 1-page UBE π , σ in which s is the first vertex of π ; see Fig. 5(b).We obtain a 2-page UBE π, σ of G, by setting π = π , σ(e) = σ (e) = 1 for each edge e = sw, and σ(sw) = 2.A symmetric argument can be used to prove the statement with respect to the vertex t.
Proof.By Lemma 14, we can assume that all the blocks of G are non-trivial, i.e., correspond to cycles.Also, let T be the BC-tree of G rooted at any block ρ.
We prove the theorem by induction on the number β(T ) of blocks in T .In fact, we prove the following slightly stronger statement: G admits a 6-page UBE in which the edges of each block lie on at most two pages.
If β(T ) = 1, then G consists of a single directed cycle, and the statement follows from [23,Lemma 2.2].
If β(T ) > 1, then T contains at least one cut-vertex c whose children blocks ν 1 , . . ., ν k are all leaves of T .Let µ be the parent block of c.Consider the subgraph of G induced by all the edges of the blocks of G different from ν 1 , . . ., ν k .Note that, by induction, G admits a 6-page UBE π , σ in which the edges of µ lie on two distinct pages, say pages 1 and 2.
We show how to extend π , σ to a 6-page book embedding π, σ of G in which the edges of each block ν i , for i = 1, . . ., k, lie on two distinct pages, namely, either pages 1 and 2, or pages 3 and 4, or pages 5 and 6.
Recall that, by Lemma 1, there exist at most two blocks among ν 1 , . . ., ν k for which c is internal.After a possible renaming of the indexes, we may assume that (i) ν 1 and ν 2 are the blocks, if any, for which c is internal; let 0 ≤ r ≤ 2 be the number of such children; (ii) ν r+1 , . . ., ν h with h ≤ k are the blocks, if any, for which c is a sink; and (iii) ν h+1 , . . ., ν k are the blocks, if any, for which c is a source.Consider the graph G induced by the edges of G and of the blocks ν i with r + 1 ≤ i ≤ k.First, we show how to extend π , σ to a 6-page UBE π , σ of G in which the edges of the blocks ν i with r + 1 ≤ i ≤ k lie on pages 1 and 2. Refer to Fig. 11.Let π = π c − • c • π c + , where either π c − or π c + may be empty (recall that each block of G is a cycle, and thus it contains at least three vertices).For each i = r + 1, . . ., h (resp.i = h + 1, . . ., k), construct a 2-page UBE π i , σ i of ν i on pages 1 and 2 in which c is the last vertex (resp.the first vertex) of π, by Lemma 15 We obtain π as follows.We set Proof.First, we construct an auxiliary graph H as follows; refer to Fig. 6.We initialize H as the union of two directed paths p 1 = (u 1 , u 2 , . . ., u k , a, b, c, d, v 1 , v 2 , . . .,v k ) and p 2 = (w 1 , w 2 , . . ., w k , e, f, g, h, z 1 , z 2 , . . ., z k ).Further, we add to H the sets of edges E 1 := {u i v i : 1 ≤ i ≤ k} and E 2 := {w i z i : 1 ≤ i ≤ k}.Then, we add the edges ae, bw 1 , dh, v k w 1 , and v k g.Clearly, H is an st-DAG with source u 1 and sink z k .Moreover, since all the vertices of H lie in a directed path from u 1 to z k their order π H is the same in any UBE of H.Note that, in π H , the edges in E 1 ∪ {ae, dh} pairwise cross, and the same holds for the edges in E 2 ∪ {ae, dh}.This implies that UBT(H) ≥ k + 2. In fact, it is immediate to see that UBT(H) = k + 2. We have the following properties for H.
We obtain G as follows.First, we initialize G to the union of G and H.Then, we add to G the edges cv and vf , for any vertex v of G. Clearly, G is an st-DAG with source u 1 and sink z k , and can be computed in O(n) time.
First, we show that γ(G ) ∈ O(k).Note that p 1 and p 2 each contain 2k + 4 vertices.Therefore, it is possible to dominate each such path by selecting k + 2 of its vertices.Further, we can dominate each vertex of G by selecting either the vertex c or the vertex f .
Next, we show that G admits a k-page UBE π, σ if and only if G admits a (k + 2)-page UBE π , σ .Suppose first that G admits a k-page UBE π, σ .We obtain a (k + 2)-page UBE π , σ as follows.Let π 1 H and π 2 H be the order of the vertices of p 1 and of p 2 in π H , respectively.We set π = π 1 H • π • π 2 H . Further, we assign the edges of G to the same (up to k) pages in σ as in σ; then, we set and (iii) σ (cv) = k + 1 and σ (vf ) = k + 2 for any vertex v of G.This concludes the construction of π , σ .To see that π , σ is indeed a (k + 2)-page UBE of G observe, in particular, that σ fulfils Property 1 and that the only edges of E(G ) \ E(G) that cross the edges of G are the edges cv and vf , where v is a vertex of G, and that such edges are assigned to the pages k + 1 and k + 2, respectively, which are not used by the edges of G.
Suppose now that G admits a (k + 2)-page UBE π , σ .We obtain a k-page UBE π, σ as follows.First, we show the following.Property 2. In π , all the vertices of G lie after v k and before w 1 .
Proof.Observe that, all the vertices of G must trivially lie between c and f , since edges cv and vf exist in G for any vertex v of G. Suppose, for a contradiction, that a vertex v of G lies after w 1 .Then, the edge cv would cross (at least) the following k+2 edges: all the edges of E 1 , the edge dh, and the edge bw 1 .However, the mentioned edges pairwise cross, which yields a contradiction.The fact that no vertex of G lies before v k can be proved analogously.
By Properties 1 and 2, we have that for each vertex v of G, it holds that σ (cv) = σ (ae) (which, w.l.o.g., we let be k + 1) and σ (vf ) = σ (dh) (which, w.l.o.g., we let be k + 2).Therefore, σ may place in pages k + 1 and k + 2 only edges connecting two consecutive vertices of G. Thus, by redefining σ (e) = 1 for any such an edge, we obtain a (k + 2)-page UBE of G in which the edges of G are assigned to k pages.Removing the vertices of H and their incident edges yields the desired k-page UBE π, σ of G.
C.2 Missing Proofs of Section 4.2 Lemma 16 ( ).G admits a τ -page UBE that can be computed in O(τ • n) time.
Proof.Let π be any topological ordering of G, that is, any ordering such that for every edge uv of G, π(u) < π(v).It is well-known that any DAG with n vertices and m edges admits at least one topological ordering, which can be computed in O(n + m) time (see, e.g., [12]).Since G has O(τ • n) edges, π can be computed in O(τ • n) time.Now consider the following page assignment σ, which again can be computed in O(τ • n) time.Let U = V \ C; for each 1 ≤ i ≤ τ , we set σ(e) = i for all edges e = (u, c i ) with u ∈ U ∪ {c 1 , . . ., c i−1 }.By construction, for each 1 ≤ i ≤ τ , all edges in page i are incident to c i , and thus no two of them cross each other.Moreover, by definition of topological ordering, for every edge uv of G, it holds π(u) < π(v).Therefore, the pair π, σ is a τ -page UBE of G and has been computed in O(τ • n) time.
Proof.Since removing a vertex u from a k-page UBE yields a h-page UBE of G − u with h ≤ k, one direction follows.
For the other direction, let u ∈ V U , such that |V U | ≥ 2 • k τ + 2. Suppose that G − u admits a k-page UBE π, σ .Two vertices u 1 , u 2 ∈ V U \ {u} are page equivalent, if for each vertex w ∈ U , the edges u 1 w and u 2 w are both assigned to the same page according to σ.By definition of type, each vertex in V U has degree exactly |U |, hence this relation partitions the vertices of V U into at most k |U | ≤ k τ sets.Since |V U \ {u}| ≥ 2 • k τ + 1, at least three vertices of this set, which we denote by u 1 , u 2 , and u 3 , are page equivalent.Consider now the graph induced by the edges of these three vertices that are assigned to a particular page.Since u 1 , u 2 , u 3 are all incident to the same set of c ≥ 1 vertices in C, such a graph is K c,3 .However, since K 2,3 is not outerplanar (and hence has no 1-page UBE), we have that c = 1.Then we can extend π by introducing u right next (or equivalently right before) to u 1 and assign each edge uw (or wu) incident to u to the same page as u 1 w (wu 1 ).Consider any edge uw (or symmetrically wu) introduced as described above.Such edge follows the curve of u 1 w and hence, since u 1 w is uncrossed in its page, uw does not cross any other edge of the same page.Moreover, since u and u 1 are of the same type, the edges uw and u 1 w are oriented consistently (either both outgoing from or incoming to w).Therefore, we obtained a k-page UBE of G, which implies that R.1 is safe.

Fig. 1 .
Fig. 1.Cases for the proof of Lemma 3. The edge uv is dashed.In all figures, the drawings are upward, hence the edge orientations are implied.

Fig. 2 .
Fig. 2. (a) Decomposing G into an appendage H1 (light blue) and a primary stouterpath G (light gray).(b) An st-fan decomposition of G with differently colored fans, and fat edges ei.(c) A UBE of G for the proof of Lemma 6.

Fig. 5 .
Fig. 5. (a) A cactus G and (b) a 2-page UBE of the red non-trivial block of G.

Fig. 6 .
Fig. 6.The graph G in the reduction of Theorem 4. The edges of the auxiliary graph H are solid.The black edges lie in k pages.

R. 1 :
If there exists a type U such that |V U | ≥ 2 • k τ + 2, then pick an arbitrary vertex u ∈ V U and set G := G − u.
{u} are page equivalent, if for each vertex w ∈ U , the edges u 1 w and u 2 w are both assigned to the same page according to σ.By definition of type, each vertex in V U has degree exactly |U |, hence this relation partitions the vertices ofV U into at most k |U | ≤ k τ sets.Since |V U \ {u}| ≥ 2 • k τ + 1,at least three vertices of this set are page equivalent.One can prove that these three vertices are incident to only one vertex in C. Then we can extend π by introducing u right next to any of these three vertices, say u 1 , and assign each edge uw incident to u to the same page as u 1 w.Theorem 6 ( ).The upward book thickness problem parameterized by the vertex cover number τ admits a kernel of size k O(τ ) .Corollary 1 ( ).Let G be an n-vertex graph with vertex cover number τ .For any k ∈ N, we can decide whether UBT(G) ≤ k in O(τ τ O(τ ) + τ • n) time.Also, within the same time complexity, we can compute a k-page UBE of G, if it exists.

Fig. 7 .
Fig. 7.The decomposition in incrementally maximal outerpaths of Fig. 3 before the update operation described in the proof of Lemma 9.
the 4 pages used by IN.The edges of G l−1 can cross only with the edges of the neighboring IN-EAST fan or with the edges of EAST.Since the IN-EAST fans use different pages, we only need to argue about the edges of EAST.Such edges only cross edges of G l−1 that are incident to v, hence, as in CASE 1, such edges belong to IN by the maximality of B. Thus, we can use a set of 4 pages for the edges of EAST different from the one used by IN.The edges of WEST only cross one neighboring EAST-WEST fan and one neighboring OUT-WEST fan, as well as the edges of EAST.Hence, we can use the same 4 pages used by IN.Similarly, the OUT edges only cross the edges of one neighboring OUT-WEST fan, the edges of EAST, and the edges of WEST.Thus we can use the last set of 4 pages different from those used by IN (which are the same used by WEST), by EAST, and by the fans.Overall, the computed UBE still uses at most 16 pages.B.2 Missing Proofs of Section 3.2 Lemma 13 ( ).Every biconnected st-outerplanar graph G admits a 4-page UBE.
, we obtain σ by setting σ (e) = σ (e), for any edge of G , and by setting σ (e) = σ i (e), for any edge of ν i with r + 1 ≤ i ≤ k.By construction, no two edges belonging to distinct blocks among ν r+1 , . . ., ν k intersect.Therefore, since the edges these C Missing Proofs of Section 4 C.1 Missing Proofs of Section 4.1 Theorem 4 ( ).Let G be an n-vertex DAG and let k be a positive integer.It is possible to construct in O(n) time an st-DAG G with γ(G ) ∈ O(k) such that UBT(G) ≤ k if and only if UBT(G ) = k + 2.

Theorem 6 (
).The upward book thickness problem parameterized by the vertex cover number τ admits a kernel of size k O(τ ) .Proof.Let G * be the graph obtained by applying R.1 exhaustively.We have already seen that there are at most 2 2τ distinct types.After the application of R.1, each type U is such that |V U | ≤ 2•k τ +1 vertices, and therefore G * contains at most (2• k τ + 1) • 2 2τ + τ ≤ k O(τ ) vertices and at most τ • k O(τ ) edges.Corollary 1 ( ).Let G be an n-vertex graph with vertex cover number τ .For any k ∈ N, we can decide whether UBT(G) ≤ k in O(τ τ O(τ ) + τ • n) time.Also, within the same time complexity, we can compute a k-page UBE of G, if it exists.Proof.If k ≥ τ , by Lemma 16, we can immediately return a positive answer and compute a k-page UBE in O(τ • n) time.So assume k < τ .By[10], we compute a vertex cover C of G of size τ in O(2 τ + τ • n) time.We have seen that computing the kernel G * of G can be done in O(2 O(τ ) + τ • n) time.Since we have 2 2τ types, and each of the 2 • k τ + 2 elements of the same type are equivalent in the book embedding (the position of two elements of the same type can be exchanged in a linear order without affecting the page assignment), the number of linear orders is(2 2τ ) O(k τ ) = 2 O(τ τ +1 ) .Since G * contains τ • k O(τ ) = τ O(τ )edges, the number of page assignments is then k τ O(τ ) .Overall, the book thickness of G * can be determined in 2O(τ τ +1 ) • k τ O(τ ) = k τ O(τ ) = τ τ O(τ ) time.If G * admits a k-page UBE, in O(τ • n) time,we can reinsert the vertices in G \ G * as in the proof of Lemma 17, thus obtaining a k-page UBE of G in overall O(τ τ O(τ ) +τ • n) time.