The Book Embedding Problem from a SATSolving Perspective
Abstract
In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges of the same page do not cross. In this paper, we approach the problem of determining whether a graph can be embedded in a book of a certain number of pages from a different perspective: We propose a simple and quite intuitive SAT formulation, which is robust enough to solve nontrivial instances of the problem in reasonable time. As a byproduct, we show a lower bound of 4 on the page number of 1planar graphs.
Keywords
Planar Graph Delaunay Triangulation Conjunctive Normal Form Page Number Outerplanar Graph1 Introduction
Embedding graphs in books is a fundamental issue in graph theory that has received considerable attention (see, e.g., [5] for an overview). In a book embedding [26], the vertices of a graph are restricted to a line, referred to as the spine of the book, and the edges are drawn at different halfplanes delimited by the spine, called pages of the book. The task is to find a socalled linear order of the vertices along the spine and an assignment of the edges of the graph to the pages of the book, so that no two edges of the same page cross; see Fig. 1b. The book thickness or page number of a graph is the smallest number of pages that are required by any book embedding of the graph.
Problems on book embeddings are mainly classified into two categories based on whether the graph to be embedded is planar or not. For nonplanar graphs, it is known that there exist graphs on n vertices that have book thickness \(\varTheta (n)\), e.g., the book thickness of the complete graph \(K_n\) is \(\lceil n/2 \rceil \) [3]. Sublinear book thickness have, e.g., graphs with subquadratic number of edges [24], subquadratic genus [23] or sublinear treewidth [13]. Constant book thickness have, e.g., all minorclosed graphs [6] or the ktrees for fixed k [16]. Another class of nonplanar graphs that was recently proved to have constant book thickness is the class of 1planar graphs [2].
For planar graphs, a remarkable result is due to Yannakakis, who back in 1986 proved that any planar graph can be embedded in a book with four pages [33]. However, more restricted subclasses of planar graphs allow embeddings in books with fewer pages. Bernhart and Kainen [3] showed that the graphs which can be embedded in singlepage books are the outerplanar graphs, while the graphs which can be embedded in books with two pages are the subhamiltonian ones.
It is known that not all planar graphs are subhamiltonian and the corresponding decision problem whether a maximal planar graph is Hamiltonian (and therefore twopage book embeddable) is NPcomplete [32]. However, several subclasses of planar graphs are known to be Hamiltonian or subhamiltonian, see, e.g., [1, 11, 12, 19, 21, 25].
A wellknown nonsubhamiltonian graph is the GoldnerHarary one [17]. This graph, however, is a planar 3tree and hence 3page book embeddable [18]. To the best of our knowledge, there is no planar graph whose page number is four. In other words, it is not known whether the upper bound of four pages of Yannakakis [33] is tight or not.
Our Contribution. We suggest an alternative approach to the problem of determining whether a graph can be embedded in a book of a certain number of pages. We propose a formulation of the problem as a SAT instance, which can be useful in practice (note that, apart from their independent theoretical interest, book embeddings find applications in several contexts, such as VLSI design, faulttolerant processing, sorting networks and parallel matrix multiplication, see e.g., [11, 20, 28, 30]). It turns out that our formulation is of a simple nature, quite intuitive and easytoimplement, but simultaneously robust enough to solve nontrivial instances of the problem in reasonable amount of time (e.g., within 20 min we can test whether a maximal planar graphs with up to 400 vertices is 3page book embeddable), as we will see in our experimental study.
Note that SAT formulations are not so common in graph drawing. A few notable exceptions are [4, 10, 15]. In our context, of interest is the work of Biedl et al. [4], who proposed ILP and SAT formulations for several gridbased graph problems. Their general formulation can be extended to solve our problem as well. However, from the authors’ experimental evaluation (and we could also confirm it) it follows that their approach is limited to solve relatively small instances within reasonable time, e.g., within 20 min one can cope with graphs whose size in vertices and edges does not exceed 100.

H1: There is a (maximal) planar graph whose book thickness is four.

H2: There is a 1planar graph whose book thickness is (at least) four.

H3: There is a (maximal) planar graph, which cannot be embedded in a book of three pages, if the subgraphs embedded at each page must be acyclic.

H4: There is a maximal planar graph, say \(G_a\), which in any of its book embeddings on three pages has at least one face whose edges are on the same page.

H5: There is a maximal planar graph, say \(G_c\), which in any of its book embeddings on three pages has a face, say \(f^*_c\), whose edges cannot be on the same page.
Summary and Discussion. Clearly, our ultimate goal was to find a planar graph supporting Hypothesis 1. During our extensive practical analysis, we tested several hundreds maximal planar graphs (both randomly created and crafted), but we did not manage to find one supporting Hypothesis 1. We also tested a specific graph with roughly 600 vertices out of the family of planar graphs that Yannakakis proposed to require page number four, but it turned out to be 3page book embeddable for this particular size.
We were surprised that we did not succeed in proving that Hypothesis 3 holds. Note that it is very natural to try to embed a treestructured subgraph at each of the available pages of the book, if one seeks to prove that indeed all planar graphs can be embedded in books of three pages. For example, Heath [18] who constructively proved that all planar 3trees fit into books with three pages used exactly this approach: the subgraphs embedded at each of the three pages of the book are acyclic.
Note that we managed to prove a weaker version of Hypothesis 3, according to which the input maximal planar graph has n vertices and cannot be embedded in a book with three pages, so that:(i)the subgraph assigned to each of the three pages is a tree on \(n1\) vertices and, additionally, (ii) the three vertices, that are not spanned by the three trees are pairwise adjacent forming a face \(f_o\) of the graph, say w.l.o.g. its outerface. This negative results implies that it is not always possible to construct a 3page book embedding based on a Schnyder decomposition into three trees (regardless of the linear order underneath).
From our experimental evaluation (see Sect. 3), we quickly observed that the practical limitation of testing the bookembedability of maximal planar graphs on three pages with our SAT formulation (that we present in Sect. 2) lies at around 600 to 700 vertices. Larger graphs lead to instance sizes, that excess several gigabytes of random access memory. Hypotheses 4 and 5 in conjunction describe an approach, that could potentially overcome this bottleneck. To see this, assume that the two planar graphs, denoted by \(G_a\) and \(G_c\) in Hypotheses 4 and 5, exist (note, however, that we did not succeed in finding them). If for each face \(f_a\) of \(G_a\), we create a copy of graph \(G_c\) and identify each of the vertices of face \(f^*_c\) of \(G_c\) with one of the vertices of face \(f_a\) of \(G_a\), then we will obtain a (drastically larger) planar graph, that is not 3page bookembeddable. This is because \(G_a\) must contain at least one face whose edges are on the same page, while in the same time face \(f^*_c\) would require at least one of them not to be at the same page.
2 SAT Formulation
Let \(G=(V,E)\), with \(V=\{v_1,v_2,\ldots ,v_n\}\) and \(E=\{e_1,e_2,\ldots ,e_m\}\), be a graph for which we seek to decide, whether it can be embedded in a book with \(p \ge 2\) pages. Next we describe a logic formula \(\mathcal {F}(G,p)\) that will solve this problem by encoding it as a SAT instance. Recall that any SAT problem can be described in conjunctive normal form (CNF), which is a conjunction of clauses; each clause being a disjunction of (possibly negated) literals. We will define \(\mathcal {F}(G,p)\) by its set of variables and a corresponding set of rules. The rules will ensure the proper assignment of the variables and will be given in propositional logic, which can be converted into CNF clauses straightforwardly [27].
Theorem 1
Let \(G = (V,E)\) be a graph and \(p \in \mathbb {N}\). Then, G admits a book embedding on p pages, if and only if, \(\mathcal {F}(G,p)\) is satisfiable. In addition, \(\mathcal {F}(G,p)\) has \(O(n^2+m^2+pm)\) variables and \(O(n^3+m^2)\) clauses.
Proof
The number of \(\sigma \), \(\chi \) and \(\phi \)variables are \(O(n^2)\), \(O(m^2)\) and O(pm), respectively, which implies that \(\mathcal {F}(G,p)\) has \(O(n^2+m^2+pm)\) variables. The number of clauses of \(\mathcal {F}(G,p)\) is dominated by the number of transitivity, samepage and planarity rules, which yield in total \(O(n^3+m^2)\) clauses. So, to prove this theorem, it remains to show that: (i) a book embedding on p pages yields a satisfying assignment of \(\mathcal {F}(G,p)\) and (ii)a satisfying assignment of \(\mathcal {F}(G,p)\) yields a book embedding on p pages.

The transitivity and asymmetry rules are satisfied by (\(\hat{\sigma }, \hat{\phi }, \hat{\chi }\)), since \(\hat{\sigma }\) is a complete order over the vertices of G (by definition of the assignment).

The direction rules are also satisfied, since we can assume w.l.o.g. that in \(\mathcal {E}(G,p)\) vertex \(v_1 \in V\) is the first vertex along the spine and \(v_2\) is to the left of \(v_3\). Note that if this is not the case, then we can circularlyshift the vertices of G along the spine and potentially mirror \(\mathcal {E}(G,p)\) and obtain an equivalent embedding which has the aforementioned properties; see e.g., [33].

The page assignment rule is trivially satisfied by the definition of the assignment and the fact that \(\mathcal {E}(G,p)\) was given.

The fixed page assignment rule can be satisfied as well, since we can assume w.l.o.g. that the first page of \(\mathcal {E}(G,p)\) is the page where edge \(e_1 \in E\) is assigned to.

The same page rule is trivially satisfied due to the definition of the assignment.

It remains to show that all planarity rules are satisfied. For the sake of contradiction, assume that the assignment (\(\hat{\sigma }, \hat{\phi }, \hat{\chi }\)) violates a planarity rule for some pair of edges \( (v_i,v_j)\) and \((v_k, v_\ell )\). We know that \(\hat{\chi }((v_i,v_j), (v_k, v_\ell )) = \texttt {true}\) and further \(\hat{\sigma }(v_i,v_k) = \hat{\sigma }(v_k,v_j) = \hat{\sigma }(v_j,v_\ell ) = \texttt {true}\). Hence, in \(\mathcal {E}(G,p)\) we have that \(v_k\) is between \(v_i\) and \(v_j\), while \(v_\ell \) is not between \(v_i\) and \(v_j\). Thus, \((v_i,v_j)\) and \((v_k, v_\ell )\) are on the same page and cross in \(\mathcal {E}(G,p)\), which is a clear contradiction.
So far, we have described a SAT formulation that tests, whether a given graph \(G=(V,E)\) admits an embedding in a book with \(p \ge 2\) pages. Of course, this formulation can be extended with additional variables and rules. In the following, we will introduce three different extensions, which encode Hypotheses 3, 4 and 5.
2.1 A First Variant to check Hypothesis 3
In this subsection, we present a SAT formulation to check Hypothesis 3. Recall that, we seek to check whether a maximal planar graph G can be embedded in \(p=3\) pages, so that the subgraph assigned to each of the three pages is an acyclic graph. In the following, we will extend formula \(\mathcal {F}(G,3)\) with new variables and rules to encode the additional requirement. We denote by \(\mathcal {F}_f(G,3)\) the resulting formula.
Theorem 2
Let \(G = (V,E)\) be a (maximal) planar graph. Then, G admits a book embedding on three pages, so that the subgraph assigned to each of the three pages is a forest, if and only, if \(\mathcal {F}_f(G,3)\) is satisfiable.
Proof
We use the same technique as in the proof of Theorem 1. So, consider an embedding \(\mathcal {E}(G,3)\) in three pages yield by our formulation. We claim that the subgraphs embedded at each page are acyclic. For contradiction, assume that there is a cycle \(\mathcal {C}_q\) at page q. If we direct each edge of \(\mathcal {C}_q\) from the child to the parent vertex, then all edges of \(\mathcal {C}_q\) must have the same orientation, that is, either clockwise or counterclockwise along \(\mathcal {C}_q\) (otherwise, there is a vertex of \(\mathcal {C}_q\) that has two parents, deviating the single parent rule). The transitivity of the ancestor relationship implies that the antisymmetry property is deviated along \(\mathcal {C}_q\), which is a contradiction. Hence, the subgraphs embedded at each page of \(\mathcal {E}(G,3)\) are indeed acyclic. Following similar arguments as in the second part of the proof of Theorem 1, we can easily prove that a satisfying assignment of \(\mathcal {F}_f(G,3)\) yields a book embedding on 3 pages, in which the subgraph assigned to each page is a forest, which completes the proof. \(\square \)
2.2 A Second Variant to Check Hypothesis 4
Assume that \(G_a=(V_a, E_a)\) is a maximal planar graph, that is embeddable in a book with 3 pages. Let \(\varDelta (G_a) = \{ f_1,f_2, \ldots ,f_{2\vert V_a \vert 4}\}\) be the set of faces of \(G_a\). In the following, we describe an extension to the formula \(\mathcal {F}(G_a,3)\) that forbids the socalled unicolored faces, that is, faces whose edges are assigned to the same page of the book. We denote the resulting formula by \(\mathcal {F}_a(G_a,3)\).
Theorem 3
\(\mathcal {F}_a(G_a,3)\) is unsatisfiable, if and only if, for every possible book embedding \(\mathcal {E}(G_a,3)\) there exists a unicolored face \(f_i \in \varDelta (G_a)\), \(i=1,\ldots ,2V_a4\).
Proof
Directly follows from the validity of \(\mathcal {F}(G_a,3)\). \(\square \)
2.3 A Third Variant to Check Hypothesis 5
Theorem 4
\(\mathcal {F}_{c}(G_c;f^*,3)\) is unsatisfiable, if and only, if there exists no book embedding of \(G_c\) with \(f^*\) being unicolored.
Proof
Directly follows from the validity of \(\mathcal {F}(G_c,3)\). \(\square \)
3 Experiments
In this section, we present an experimental evaluation of our SAT formulation. We ran our experiments on a Linux machine with four cores at 2, 5 GHz and 8 GB of RAM. The implementation that creates the SAT instances was done in Java. For solving the SAT instances, we used the SApperloT solver [22]. This solver is as fast as the wellknown minisat [14] solver for smaller graphs, but it considerably outperforms minisat for increasing instance sizes. The runtime we report consists of both, the time to create the instance and the time to solve it. Note that the time to create the SAT instance for small graphs is neglectable. For large graphs, however, that step can take a few minutes.
Established Benchmark Sets. Since the Rome and the North graphs are popular test sets for planar and nearly planar graphs, we also used them as test sets for our experiment (cf. http://www.graphdrawing.org). The Rome graphs are 11534 graphs; 3281 of them are planar and 8253 are nonplanar. Their average density is 0.069, where the density of a graph \(G=(V,E)\) is \(2E/(V(V1))\). The number of vertices of the Rome graphs range from 10 to 110. The corresponding number of edges range from 9 to 158.
It is eyecatching, that all planar Rome graphs are 2page book embeddable (see Table 1). The nonplanar ones are 3page embeddable. But since the Rome graphs are very sparse this result was more or less expected. Note that \(99\,\%\) of the planar Rome graphs (that is, 3248 out of 3282) are solved within 2 s. For the nonplanar Rome graphs, the same ratio (that is, 8169 out of 8253) is achieved after 6.25 s.
Overview of the results for the established benchmark sets.
planar  nonplanar  

Graph class  \(\#\)  \(p=2\)  \(\#\)  \(p=3\)  \(p=4\)  \(p=5\)  see below 
Rome  3281  3281  8253  8253  0  0  
North  854  854  423  329  25  8  61 
Crafted Graphs. To prove Hypothesis 1, we also crafted several maximal planar graphs with at least 500 vertices each, which we tested for 3page book embeddability. To avoid testing Hamiltonian graphs, we adopted a twostep approach that was inspired by the graph class that Yannakakis proposed as candidate to require four pages. In the first step, we chose a triangulated planar (not necessarily nonHamiltonian) graph as the base for the second step. In the second step, we augmented the base graph by specific operations to make it nonHamiltonian (and therefore not 2page embeddable). Examples of these operations are: (i) stellate a face f, that is, introduce a new vertex and connect it to all vertices of f, (ii) replace a triangular face by an octahedron, (iii) embed a nonHamiltonian planar graph \(G_f\) to a face f by identifying the vertices of f with the vertices of a particular face of \(G_f\).
We observed that these operations most of the times yield nonHamiltonian planar graphs. Note that they do not generate the whole class of nonHamiltonian planar graphs and not even a uniformlydistributed random subset. The graphs, that we crafted and tested with this approach, were all maximal planar with at least 500 and at most 700 vertices. The runtime to check each instance ranged from few hours to a couple of days.
1Planar Graphs. To check Hypothesis 2 for more than four pages, we initially generated all 2,098,675 planar triconnected quadrangulations with 25 vertices and minimum degree three using plantri [8]. By augmenting every face with two crossing edges, the generated quadrangulations yield optimal 1planar graphs (recall that a 1planar graph on n vertices is said to be optimal, if it has exactly \(4n8\) edges, which is the maximum possible [7]). Our experiments showed that all tested optimal 1planar graphs required four pages. The runtime distribution is shown in Fig. 2c. Computing a 4page embedding was always fast: For \(99.06\,\%\) of the graphs the solver found a solution within 4.7 s. The maximum runtime for a single instance was 186 s. Proving that no 3page embedding existed was harder. In less than 5 min., \(94.4\,\%\) of the instances could be solved. However, for very few instances this could take up to two hours.
To obtain a better understanding of the connection between the runtime of our approach and the size of the graphs, we randomly created 8312 optimal 1planar graphs of different sizes varying from 50 to 155 vertices. Starting from the cube graph (see Fig. 1), we iteratively applied at random one of the two operations described in [29] in order to generate all optimal 1planar graphs, until we reached the desired size of the graph. The runtime to compute 4page embeddings for these graphs is shown in Fig. 2d. For nearly all graphs up to 100 vertices a 4page embedding could be computed within two minutes. However, with increasing vertexcount, the amount of graphs that could take up to several hours to compute a 4page embedding rises rapidly.
Randomized Planar Graphs. To check Hypotheses 35, we generated a large set of random maximal planar graphs as follows. We applied Delaunay triangulation on a set of randomly created points within a triangular region. To avoid Hamiltonian graphs, we stellated every face of the produced Delaunay triangulations. Our results are summarized in the following. Note that none of the tested graphs corroborate Hypotheses 35.

For Hypothesis 3, we tested 15, 040 maximal planar graphs of varying number of vertices between 25 and 80. We were able to solve \(70.78\,\%\) of the instances (that is, 10, 646 out of 15, 040) within 3 min. and \(76.37\,\%\) (that is, 11, 487 out of 15, 040) of the instances within 20 min., which was the time limit of the computation.

For Hypothesis 4, we tested 7, 174 maximal planar graphs of varying number of vertices between 59 and 125. We were able to solve \(92.75\,\%\) of the instances (that is, 6621 out of 7174) within 10 min. and \(99\,\%\) of the instances within an hour. The maximum time that was needed to solve a single instance was 5 h and 6 min.

For Hypothesis 5, we managed to test 277, 284 maximal planar graphs of varying number n of vertices between 59 and 95. Every single instance, each containing \(2n4\) different SAT formulas, required only few seconds to be tested.
4 Conclusions and Discussion

H6: All planar graphs admit 3page book embeddings, in which additionally the subgraphs assigned to each page are trees.

H7: Optimal 1planar graphs have book thickness four.

H8: The book thickness of general 1planar graphs is not more than five.
Our experimental evaluation showed that our approach can be useful, since it can cope with nontrivial instances in reasonable amount of time. However, around the optimal solution, where the problem switches from unsatisfiable to satisfiable, we observed the wellknown phase transitional behavior for SAT problems [9]. So, in this context, we mention two more major open problems. The first one is to refine our SAT approach to support denser graphs. The second one is to extend it to other gridbased problems. For example, with our formulation it could be possible to solve larger instances for several of the problems studied, e.g., by Biedl et al. [4].
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