arXiv subjects

Michael A. Bekos

Publications and source records attributed to Michael A. Bekos.

64 records · Page 4Linked to original sources

arXiv2015

The Book Thickness of 1-Planar Graphs is Constant

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 on the same page do not cross. In this paper, we prove that every $1$-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is $O(\sqrt{n})$, where $n$ is the number of vertices of the graph.

cs.DS↗

arXiv2014

Improved Approximation Algorithms for Box Contact Representations

We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called \textsc{Contact Representation of Word Networks} (\textsc{Crown}) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. \textsc{Crown} is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, \textsc{Max-Crown}, in which realizing each desired adjacency yields a certain profit. We present the first $O(1)$-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.

cs.DS↗

arXiv2014

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm to test whether a given graph is maximal outer-fan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-hard, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given.

cs.CG↗

arXiv2014

Planar Octilinear Drawings with One Bend Per Edge

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A $k$-planar graph is a planar graph in which each vertex has degree less or equal to $k$. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size $O(n^2) \times O(n)$. For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge.

cs.DS↗

arXiv2014

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.

cs.DM↗

arXiv2013

Smooth Orthogonal Drawings of Planar Graphs

In \emph{smooth orthogonal layouts} of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axis-aligned tangents. In this paper, we study the problem of finding smooth orthogonal layouts of low \emph{edge complexity}, that is, with few segments per edge. We say that a graph has \emph{smooth complexity} k---for short, an SC_k-layout---if it admits a smooth orthogonal drawing of edge complexity at most $k$. Our main result is that every 4-planar graph has an SC_2-layout. While our drawings may have super-polynomial area, we show that, for 3-planar graphs, cubic area suffices. Further, we show that every biconnected 4-outerplane graph admits an SC_1-layout. On the negative side, we demonstrate an infinite family of biconnected 4-planar graphs that requires exponential area for an SC_1-layout. Finally, we present an infinite family of biconnected 4-planar graphs that does not admit an SC_1-layout.

cs.CG↗

arXiv2013

Many-to-One Boundary Labeling with Backbones

In this paper we study \emph{many-to-one boundary labeling with backbone leaders}. In this new many-to-one model, a horizontal backbone reaches out of each label into the feature-enclosing rectangle. Feature points that need to be connected to this label are linked via vertical line segments to the backbone. We present dynamic programming algorithms for label number and total leader length minimization of crossing-free backbone labelings. When crossings are allowed, we aim to obtain solutions with the minimum number of crossings. This can be achieved efficiently in the case of fixed label order, however, in the case of flexible label order we show that minimizing the number of leader crossings is NP-hard.

cs.CG↗

arXiv2011

Geometric Simultaneous RAC Drawings of Graphs

In this paper, we introduce and study "geometric simultaneous RAC drawing problems", i.e., a combination of problems on geometric RAC drawings and geometric simultaneous graph drawings. To the best of our knowledge, this is the first time where such a combination is attempted.

cs.DS↗

arXiv2010

Maximizing the Total Resolution of Graphs

A major factor affecting the readability of a graph drawing is its resolution. In the graph drawing literature, the resolution of a drawing is either measured based on the angles formed by consecutive edges incident to a common node (angular resolution) or by the angles formed at edge crossings (crossing resolution). In this paper, we evaluate both by introducing the notion of "total resolution", that is, the minimum of the angular and crossing resolution. To the best of our knowledge, this is the first time where the problem of maximizing the total resolution of a drawing is studied. The main contribution of the paper consists of drawings of asymptotically optimal total resolution for complete graphs (circular drawings) and for complete bipartite graphs (2-layered drawings). In addition, we present and experimentally evaluate a force-directed based algorithm that constructs drawings of large total resolution.

cs.DS↗

arXiv2010

The Straight-Line RAC Drawing Problem is NP-Hard

Recent cognitive experiments have shown that the negative impact of an edge crossing on the human understanding of a graph drawing, tends to be eliminated in the case where the crossing angles are greater than 70 degrees. This motivated the study of RAC drawings, in which every pair of crossing edges intersects at right angle. In this work, we demonstrate a class of graphs with unique RAC combinatorial embedding and we employ members of this class in order to show that it is NP-hard to decide whether a graph admits a straight-line RAC drawing.

cs.DS↗