Two-Page Book Embeddings of 4-Planar Graphs

Back in the Eighties, Heath showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book-embedding viewpoint of the problem.Comment: 21 pages, 16 Figures. A shorter version is to appear at STACS 201

On Optimal 2- and 3-Planar Graphs

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized, this has not been the case for $k \geq 2$. For $k=2,3$ and $4$, upper bounds on the edge density have been developed for the case of simple graphs by Pach and T\'oth, Pach et al. and Ackerman, which have been used to improve the well-known "Crossing Lemma". Recently, we proved that these bounds also apply to non-simple $2$- and $3$-planar graphs without homotopic parallel edges and self-loops. In this paper, we completely characterize optimal $2$- and $3$-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes

On a conjecture of Lovász on circle-representations of simple 4-regular planar graphs

Lovász conjectured that every connected 4-regular planar graph $G$ admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of $G$ correspond to the intersection and touching points of the circles and the edges of $G$ are the arc segments among pairs of intersection and touching points of the circles. In this paper, we settle this conjecture. In particular, (a) we first provide tight upper and lower bounds on the number of circles needed in a realization of any simple 4-regular planar graph, (b) we affirmatively answer Lovász's conjecture, if $G$ is 3-connected, and, (c) we demonstrate an infinite class of simple connected 4-regular planar graphs which are not 3-connected (i.e., either simply connected or biconnected) and do not admit realizations as a system of circles

Two-Page Book Embeddings of 4-Planar Graphs (Extended Draft Version)

Back in the Eighties, Heath showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book-embedding viewpoint of the problem

Two-Page Book Embeddings of 4-Planar Graphs

Back in the eighties, Heath [Algorithms for embedding graphs in books. PhD thesis, University of North Carolina, Chapel Hill, 1985] showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book embedding viewpoint of the problem

Parameterized Algorithms for Upward Planarity

We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, single-source digraphs and outerplanar digraphs

Perfect Smooth Orthogonal Drawings

Smooth orthogonal drawings were recently intro- duced with the view of combining two different graph drawing approaches: Orthogonal drawings and Lombardi drawings. In this paper, we focus on perfect smooth orthogonal drawings in which each edge is made of either a rectilinear segment or a circular arc. We prove that every 3-planar graph admits a planar perfect smooth orthogonal drawing. If we relax planarity constraints, we show that every graph of maximum degree 4 admits a (non-planar) perfect smooth orthogonal drawing. We demonstrate that there exist infinitely many planar graphs that do not admit planar perfect smooth orthogonal drawings under the Kandinsky model. Finally, we introduce classes of graphs admitting perfect smooth orthogonal drawings of different styles and study relations between these classes