Michael A. Bekos

Michael A. Bekos
University of Ioannina | UOI · Department of Mathematics

Mathematics

About

162
Publications
38,351
Reads
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1,030
Citations
Citations since 2017
103 Research Items
628 Citations
2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
Additional affiliations
February 2022 - present
University of Ioannina
Position
  • Professor (Assistant)
December 2012 - January 2022
University of Tuebingen
Position
  • PostDoc Position
Description
  • Drawing graphs with high resolution and applications in mobile devices
Education
September 2004 - November 2008
National Technical University of Athens
Field of study
  • Ph.D.: Computer Science and Discrete Mathematics
September 2002 - September 2004
University of Ioannina
Field of study
  • M.Sc.: Computational Mathematics and Computer Science
September 1998 - September 2002
University of Ioannina
Field of study
  • B.Sc.: Mathematics

Publications

Publications (162)
Preprint
Full-text available
Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines (layers), and their relationships (edges) are represented by segments connecting vertices. Methods for constr...
Chapter
The page-number of a directed acyclic graph (a DAG, for short) is the minimum k for which the DAG has a topological order and a k-coloring of its edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological order. In 1999, Heath and Pemmaraju conjectured that the recognition of DAGs with page-number 2...
Chapter
We continue the study of linear layouts of graphs in relation to known data structures. At a high level, given a data structure, the goal is to find a linear order of the vertices of the graph and a partition of its edges into pages, such that the edges in each page follow the restriction of the given data structure in the underlying order. In this...
Chapter
Strictly-convex straight-line drawings of 3-connected planar graphs in small area form a classical research topic in Graph Drawing. Currently, the best-known area bound for such drawings is O(n2)×O(n2), as shown by Bárány and Rote by means of a sophisticated technique based on perturbing (non-strictly) convex drawings. Unfortunately, the hidden con...
Chapter
When interacting with a visualization of a bipartite graph, one of the most common tasks requires identifying the neighbors of a given vertex. In interactive visualizations, selecting a vertex of interest usually highlights the edges to its neighbors while hiding/shading the rest of the graph. If the graph is large, the highlighted subgraph may not...
Article
Full-text available
We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most w . The conjecture has been confirmed for posets of width $$w=2$$ w = 2 via so-called lazy linear extension....
Article
Full-text available
A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+1)$$\end{document}-rainb...
Preprint
Full-text available
We continue the study of linear layouts of graphs in relation to known data structures. At a high level, given a data structure, the goal is to find a linear order of the vertices of the graph and a partition of its edges into pages, such that the edges in each page follow the restriction of the given data structure in the underlying order. In this...
Article
Full-text available
A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. In 2017, Pupyrev...
Preprint
Full-text available
Strictly-convex straight-line drawings of $3$-connected planar graphs in small area form a classical research topic in Graph Drawing. Currently, the best-known area bound for such drawings is $O(n^2) \times O(n^2)$, as shown by B\'{a}r\'{a}ny and Rote by means of a sophisticated technique based on perturbing (non-strictly) convex drawings. Unfortun...
Preprint
Full-text available
The page-number of a directed acyclic graph (a DAG, for short) is the minimum $k$ for which the DAG has a topological order and a $k$-coloring of its edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological order. In 1999, Heath and Pemmaraju conjectured that the recognition of DAGs with page-numb...
Article
Full-text available
A linear layout of a graph typically consists of a total vertex order, and a partition of the edges into sets either of non-crossing edges, called stacks, or of non-nested edges, called queues. The stack (queue) number of a graph is the minimum number of required stacks (queues) in any linear layout of it. Mixed linear layouts combine these layouts...
Preprint
Full-text available
Motivated by cognitive experiments providing evidence that large crossing-angles do not impair the readability of a graph drawing, RAC (Right Angle Crossing) drawings were introduced to address the problem of producing readable representations of non-planar graphs by supporting the optimal case in which all crossings form 90{\deg} angles. In this w...
Preprint
Full-text available
A map graph is a graph admitting a representation in which vertices are nations on a spherical map and edges are shared curve segments or points between nations. We present an explicit fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. The algorithm has time complexity that is linear in the size of the graph...
Article
Full-text available
We prove that every set $\mathcal S$ of $\Delta$ slopes containing the horizontal slope is universal for $1$-bend upward planar drawings of bitonic $st$-graphs with maximum vertex degree $\Delta$, i.e., every such digraph admits a $1$-bend upward planar drawing whose edge segments use only slopes in $\mathcal S$. This result is worst-case optimal i...
Preprint
Full-text available
We continue the study of the area requirement of convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally require the obtained drawings to have bounded edge-vertex resolution, that is, the closest distance between a...
Preprint
Full-text available
Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph clas...
Article
Set systems can be visualized in various ways. An important distinction between techniques is whether the elements have a spatial location that is to be used for the visualization; for example, the elements are cities on a map. Strictly adhering to such location may severely limit the visualization and force overlay, intersections and other forms o...
Chapter
Full-text available
We continue the study of the area requirement of convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally require the obtained drawings to have bounded edge-vertex resolution, that is, the closest distance between a...
Chapter
Full-text available
A k-queue layout is a special type of a linear layout, in which the linear order avoids \((k+1)\)-rainbows, i.e., \(k+1\) independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e., the minimum value of k for which a k-queue layout is feasible. Recently, Dujmović et al. [13] showed th...
Chapter
Full-text available
Computing a morph between two drawings of a graph is a classical problem in computational geometry and graph drawing. While this problem has been widely studied in the context of planar graphs, very little is known about the existence of topology-preserving morphs for pairs of non-planar graph drawings. We make a step towards this problem by showin...
Preprint
Full-text available
A linear layout of a graph typically consists of a total vertex order, and a partition of the edges into sets of either non-crossing edges, called stacks, or non-nested edges, called queues. The stack (queue) number of a graph is the minimum number of required stacks (queues) in a linear layout. Mixed linear layouts combine these layouts by allowin...
Preprint
Full-text available
A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e., the minimum value of k for which a k-queue layout is feasible. Recently, Dujmovi\'c et al. [J. ACM, 67(4), 22:1...
Preprint
Full-text available
Computing a morph between two drawings of a graph is a classical problem in computational geometry and graph drawing. While this problem has been widely studied in the context of planar graphs, very little is known about the existence of topology-preserving morphs for pairs of non-planar graph drawings. We make a step towards this problem by showin...
Article
Full-text available
We study the algorithmic problem of computing drawings of graphs in which (i) each vertex is a disk with fixed radius ρ, (ii) each edge is a straight-line segment connecting the centers of the two disks representing its end-vertices, (iii) no two disks intersect, and (iv) the distance between an edge segment and the center of a non-incident disk, c...
Chapter
In this chapter we discuss algorithmic techniques that are frequently applied in external labeling. We split them into non-exact (Section 4.1) and exact algorithms (Section 4.2). While exact algorithms solve instances of external labeling problems and provide an optimality guarantee for the solution, non-exact approaches refrain from such guarantee...
Chapter
Straight-line leaders form the simplest way to establish the visual association between features and their external labels. Further, they can be easily traced by the reader [6]. In this chapter, we first give an overview of different aspects of labelings with straight-line leaders (Section 5.1) and then discuss important contributions (Section 5.2)...
Chapter
In this chapter, we present a unified and extensible taxonomy of different labeling models proposed in the literature. At the same time, it serves as an entry point for the interested reader who wants to get familiar with the basic concepts of external labeling. First, we formally introduce the most important terminology and concepts on external la...
Chapter
The previous chapters provided an in-depth discussion of the state of the art on external labeling algorithms. Our literature review identified that two mostly independent research communities have studied external labeling problems extensively: the visual computing community as well as the algorithms community. In visual computing the vast majorit...
Chapter
In this chapter we discuss various visual aspects that affect the aesthetic quality and usability of external labeling. In general, it is far from obvious which type of external labeling is the best choice and this highly depends on the application. We discuss several important criteria that must be taken into account when deciding on a type of ext...
Chapter
Polyline leaders form a natural generalization of straight-line leaders by supporting bends. To facilitate readability, polyline leaders usually have a schematic and tidy appearance caused by the restriction to segment orientations that are aligned with the Cartesian coordinate axes or possibly with their two bisecting diagonals. As one would expec...
Chapter
Bitonic st-orderings for st-planar graphs were recently introduced as a method to cope with several graph drawing problems. Notably, they have been used to obtain the best-known upper bound on the number of bends for upward planar polyline drawings with at most one bend per edge. For an st-planar graph that does not admit a bitonic st-ordering, one...
Chapter
A topological graphTopological graph is called k-planar, for k≥0, if each edge has at most k crossings; hence, by definition, 0-planar topological graphs are plane. An abstract graph is called k-planar if it is isomorphic to a k-planar topological graph, i.e., if it can be drawn on the plane with at most k crossings per edge. While planar and 1-pla...
Chapter
A fan-planar graph is a graph that admits a drawing, in which each edge can cross only edges with a common endvertex, and this endvertex is on the same side of the edge. Hence, by definition, fan-planar graphs extend the class of 1-planar graphs1-planar graph, but still form a proper subclass of 3-quasiplanar graphs, as they cannot contain three mu...
Article
Full-text available
A queue layout of a graph G consists of a linear order of the vertices of G and a partition of the edges of G into queues, so that no two independent edges of the same queue are nested. The queue number of graph G is defined as the minimum number of queues required by any queue layout of G. In this paper, we continue the study of the queue number o...
Preprint
Full-text available
We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most w. The conjecture has been confirmed for posets of width w=2 via so-called lazy linear extension. We extend a...
Preprint
Full-text available
A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pup...
Preprint
Full-text available
An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignment...
Preprint
Full-text available
We study the algorithmic problem of computing drawings of graphs in which (i) each vertex is a disk with constant radius r, (ii) each edge is a straight-line segment connecting the centers of the two disks representing its end-vertices, (iii) no two disks intersect, and (iv) the edge-vertex resolution is at least r, that is, no edge segment interse...
Preprint
Full-text available
An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of gra...
Article
Full-text available
A k-bend right-angle-crossing drawing (or k-bend RAC drawing, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are 90∘. Accordingly, a graph that admits a k-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for short); a 0-ben...
Preprint
Full-text available
We present a prototype online system to automate the procedure of computing different types of linear layouts of graphs under different user-specific constraints. The system consists of two main components; the client and the server sides. The client side is built upon an easy-to-use editor, which supports basic interaction with graphs, enriched wi...
Preprint
An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fund...
Chapter
Full-text available
A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pup...
Chapter
Full-text available
We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most w. The conjecture has been confirmed for posets of width w=2 via so-called lazy linear extension. We extend a...
Chapter
Full-text available
An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that “point to each other” inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignment...
Article
Full-text available
The crossing resolution of a non-planar drawing of a graph is the value of the minimum angle formed by any pair of crossing edges. Recent experiments suggest that the larger the crossing resolution is, the easier it is to read and interpret a drawing of a graph. However, maximizing the crossing resolution turns out to be an NP-hard problem in gener...
Chapter
Full-text available
Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph class for its maximum edge density, another parameter that is often considered in the literature is the size o...
Preprint
Full-text available
In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings with two and three available colors and present improved bounds on the size of the monochromatic connected compo...
Chapter
Motivated by the study of ordinal embeddings in machine learning and by the recognition of Euclidean preferences in computational social science, we study the following problem. Given a graph G, together with a set of relationships between pairs of edges, each specifying that an edge must be longer than another edge, is it possible to construct a s...
Preprint
Full-text available
Lov\'asz 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...
Article
Full-text available
A simple topological graph is k-quasiplanar (k≥2) if it contains no k pairwise crossing edges, and k-planar if no edge is crossed more than k times. In this paper, we explore the relationship between k-planarity and k-quasiplanarity to show that, for k≥2, every k-planar simple topological graph can be transformed into a (k+1)-quasiplanar simple top...
Article
Full-text available
A graph is k-planar if it can be drawn in the plane such that each edge is crossed at most k>0 times. These graphs represent a natural extension of planar graphs and they are among the most investigated families in the growing field of graph drawing beyond planarity. In this paper, we study visibility representations of k-planar graphs in three dim...
Preprint
Full-text available
A simple topological graph is $k$-quasiplanar ($k\geq 2$) if it contains no $k$ pairwise crossing edges, and $k$-planar if no edge is crossed more than $k$ times. In this paper, we explore the relationship between $k$-planarity and $k$-quasiplanarity to show that, for $k \geq 2$, every $k$-planar simple topological graph can be transformed into a $...
Preprint
Full-text available
Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph class for its maximum edge density, another parameter that is often considered in the literature is the size o...
Article
Full-text available
A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. From an application perspective, greedy drawings are especially relevant to support routing schemes in ad hoc wireless networks. The existence of greedy d...
Conference Paper
A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg states that the qu...
Article
Full-text available
We describe a set of $\Delta -1$ slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree $\Delta \geq 4$; this establishes a new upper bound of $\Delta-1$ on the 1-bend planar slope number. By universal we mean that every planar graph of degree $\Delta$ has a planar drawing with at most one bend per edge and such tha...
Article
Full-text available
In this paper we consider graphs whose edges are associated with a degree of {\em importance}, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to construct layouts of these graphs in which the readability of an edge is proportional to its importance, that...
Article
External labeling is frequently used for annotating features in graphical displays and visualizations, such as technical illustrations, anatomical drawings, or maps, with textual information. Such a labeling connects features within an illustration by thin leader lines with their labels, which are placed in the empty space surrounding the image. Ov...
Article
Full-text available
We study two variants of the well-known orthogonal graph drawing model: (1) the smooth orthogonal, and (2) the octilinear. Both models are extensions of the orthogonal one, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of maximum vertex degree 4, we analyze relationships be...
Article
Full-text available
A topological graph is a graph drawn in the plane. A topological graph is k-plane, k>0, if each edge is crossed at most k times. We study the problem of partitioning the edges of a k-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for k=1, we focus on optimal 2-plane and on optimal...
Preprint
Full-text available
External labeling is frequently used for annotating features in graphical displays and visualizations, such as technical illustrations, anatomical drawings, or maps, with textual information. Such a labeling connects features within an illustration by thin leader lines with their labels, which are placed in the empty space surrounding the image. Ov...
Preprint
A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg states that the qu...
Chapter
Full-text available
A k-bend right-angle-crossing drawing (or k-bend RAC drawing, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are \(90^\circ \). Accordingly, a graph that admits a \(k\)-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for s...
Chapter
Full-text available
A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. The existence of greedy drawings has been widely studied under different topological and geometric constraints, such as planarity, face convexity, and dra...