A Tight Subexponential-time Algorithm for Two-Page Book Embedding

TL;DR

This work studies the Two-Page Book Embedding problem, showing a subexponential-time exact algorithm with runtime $2^{O(\sqrt{n})}$ and proving ETH-tightness. It develops a single-exponential fixed-parameter algorithm parameterized by treewidth by performing dynamic programming along sphere-cut decompositions and SPQR-trees, and extends tractability to the feedback edge number with a linear kernel and a $2^{O(\sqrt{k})}$-time variant. The approach integrates a normal-form witness for subhamiltonicity, a detailed SPQR-tree-based DP framework, and a kernelization/branching scheme for many-pages, yielding strong structural results and practical algorithms for planar and near-planar graphs. Overall, the paper advances the algorithmic frontier for embedding graphs on few pages and provides a versatile framework for DP on sphere-cut decompositions that may apply to other drawing problems.

Abstract

A book embedding of a graph is a drawing that maps vertices onto a line and edges to simple pairwise non-crossing curves drawn into pages, which are half-planes bounded by that line. Two-page book embeddings, i.e., book embeddings into 2 pages, are of special importance as they are both NP-hard to compute and have specific applications. We obtain a 2^(O(\sqrt{n})) algorithm for computing a book embedding of an n-vertex graph on two pages -- a result which is asymptotically tight under the Exponential Time Hypothesis. As a key tool in our approach, we obtain a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph. We conclude by establishing the fixed-parameter tractability of computing minimum-page book embeddings when parameterized by the feedback edge number, settling an open question arising from previous work on the problem.

Figures (9)

• Figure 1: A drawing of a subhamiltonian graph $G$, made of the full-edges, which is completed by the dashed edges to one of its Hamiltonian supergraphs $G_H$ (left) and the same graph drawn as a two-page book embedding (right). In both drawings the Hamiltonian cycle $H$ is colored in blue and the edges belonging to page 1 and 2 are colored with green and red, respectively.Note that the partition of the edges into the pages can be obtained from a planar drawing of $G_H$ by partitioning the edges according to the two regions given by $H$.
• Figure 2: $(a)$ shows a biconnected multi-graph $G$. $(b)$ shows the SPQR-tree $\mathcal{B}$ of $G$. $(c)$ shows the skeleton of $b$, $\textsc{Sk}(b)$, where the edge $e$ that corresponds to the child (with pertinent node) $b'$ is in bold and the dashed edge represents the reference edge. Finally, $(d)$ shows $\textsc{Pe}(b')$.
• Figure 3: The cycle $H = (u_2,P_1,u_1,v_1,P_2,u_3,v_3,P_3,v_2,u_2)$ represents a Hamiltonian cycle that crosses the $uv$-curve at least three times (in $p_1$, $p_2$ and $p_3$). Thanks to Lemma \ref{['lem:crossing']}, we obtain a Hamiltonian cycle $H'= (u_2,P_1,u_1,v_3,P_3,v_2,v_1,P_2,u_3,u_2)$ that differs from $H$ only inside the face $f=(u,u_1,u_2,u_3,v,v_3,v_2,v_1)$ and crosses the $uv$-curve two fewer times than $H$ does. Finally, note that the vertices $u$ and $v$ are part of either $P_1$, $P_2$, or $P_3$.
• Figure 4: The figure shows three different types of a node in an SPQR-tree with reference edge $(s,t)$, i.e., the types shown are (from left to right): $(\{\{L\rightarrow \{l\}\},\{R\rightarrow \{r,r'\}\},\{\{l,s\},\{r,r'\}\},\{t\})$, $(\{\{L\rightarrow \{l,l'\}\},\{R\rightarrow \{r,r'\}\},\{\{l,s\},\{l',r\},\{t,r'\}\},\emptyset)$, and $(\{\{L\rightarrow \{l\}\},\{R\rightarrow \{r\}\}, \{\{l,r\}\},\{t\})$. The subset of $\{l,l'\}$ and $\{r,r'\}$ that appears corresponds to $\psi(L)$ and $\psi(R)$ respectively. The blue edges correspond to the matching $M$ and the blue vertices corresponds to $S$.
• Figure 5: An illustration of the relationship of the parent noose $O_{a_{P}}$ and the child nooses $O_{a_{L}}$ and $O_{a_{R}}$. The illustration represents the case of Lemma \ref{['lem:triangles']} where $O'=O_{a_P} \oplus\xspace O_{a_L} \oplus\xspace O_{a_R}$ consists of two disjoint weak nooses (triangles) $O_1$ and $O_2$. Let $c_{i,j}$ be a curve between $u_i$, $u_j$ then all nooses are defined as follows: $O_{a_P} = \{c_{2,3}, c_{3,4}, c_{4,6}, c_{6,7}, c_{7,8}, c_{2,8}\}$, $O_{a_L} = \{c_{1,2},c_{2,3}, c_{3,4},c_{4,5}, c_{1,5}\}$, $O_{a_R} = \{c_{1,5},c_{5,6}, c_{6,7}, c_{7,8}, c_{1,8}\}$, $O_{1} = \{c_{1,2},c_{2,8}, c_{1,8}\}$, $O_{2} = \{c_{4,5},c_{5,6}, c_{4,6}\}$, $O'= O_1 \cup O_2$.

Theorems & Definitions (84)

• proof
• Lemma 5: GutwengerM00
• Lemma 7: DBLP:journals/jct/RobertsonS91
• Lemma 8: DornPBF10
• Lemma 10
• Lemma 12
• proof
• Lemma 13
• proof
• Lemma 14