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The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c= 1/100$, which then has gradually been improved. Here, the bounds for the density of $k$-planar graphs played a central role. Our new insight is that for $k=2,3$ the $k$-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of $k$-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number $\text{cr}(G)$ of a given graph $G$. In particular, we achieve a bound of $\text{cr}(G) \ge \frac{37}{9}m-\frac{155}{9}(n-2)$ for the range of $5n < m \le 6n$, while our second bound $\text{cr}(G) \ge 5m - \frac{203}{9}(n-2)$ is even stronger for larger $m>6n$. For $m > 6.77n$, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of $c>1/27.48$ in the Crossing Lemma. Note that the previous constant was $1/29$. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.

We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are $3$-cycles, $4$-cycles or both of them (i.e., girth $\ge 5$). For all three settings and all $k\in\{1,2,3\}$, we present lower and upper bounds on the maximum number of edges in any $k$-planar graph on $n$ vertices. Our bounds are of the form $c\,n$, for some explicit constant $c$ that depends on $k$ and on the setting. For general $k \geq 4$ our bounds are of the form $c\sqrt{k}n$, for some explicit constant $c$. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of $2$-- and $3$-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma., Comment: Appears in the Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

We show that every planar graph has a monotone topological 2-page book embedding where at most (4n-10)/5 (of potentially 3n-6) edges cross the spine, and every edge crosses the spine at most once; such an edge is called a biarc. We can also guarantee that all edges that cross the spine cross it in the same direction (e.g., from bottom to top). For planar 3-trees we can further improve the bound to (3n-9)/4, and for so-called Kleetopes we obtain a bound of at most (n-8)/3 edges that cross the spine. The bound for Kleetopes is tight, even if the drawing is not required to be monotone. A Kleetope is a plane triangulation that is derived from another plane triangulation T by inserting a new vertex v_f into each face f of T and then connecting v_f to the three vertices of f., Comment: Appears in the Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with $n$ vertices and $m$ edges can be computed in $2^m \cdot n^{O(1)}$ time. An $O(\log n)$-approximation of this number can be computed efficiently. We can decide in $2^{O(d \sqrt{k} \log (d+k))} \cdot n^{O(1)}$ time whether it suffices to delete $k$ edges of an ordered graph to obtain a $d$-planar layout (where every edge crosses at most $d$ other edges) on one page. As an additional parameter, we consider the size $h$ of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For $h=1$, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number $p$. For $h>1$, we give an XP algorithm with respect to $h+p$. Finally, we consider spine+$t$-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of $t$ tracks, each of which is a straight line on a separate page, parallel to the spine., Comment: Appears in Proc. 19th Scandinavian Symposium on Algorithm Theory (SWAT 2024)

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing several applications: we prove tight upper bounds on the edge density of various beyond-planar graph classes, including so-called $k$-planar graphs with $k=1,2$, fan-crossing / fan-planar graphs, $k$-bend RAC-graphs with $k=0,1,2$, quasiplanar graphs, and $k^+$-real face graphs. In some cases ($1$-bend and $2$-bend RAC-graphs and fan-crossing / fan-planar graphs), we thereby obtain the first tight upper bounds on the edge density of the respective graph classes. In other cases, we give new streamlined and significantly shorter proofs for bounds that were already known in the literature. Thanks to the Density Formula, all of our proofs are mostly elementary counting and mostly circumvent the typical intricate case analysis found in earlier proofs. Further, in some cases (simple and non-homotopic quasiplanar graphs), our alternative proofs using the Density Formula lead to the first tight lower bound examples., Comment: To appear in the proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the $k$-planar drawings $(k \geq 1)$, where each edge cannot cross more than $k$ times. We generalize $k$-planar drawings, by introducing the new family of min-$k$-planar drawings. In a min-$k$-planar drawing edges can cross an arbitrary number of times, but for any two crossing edges, one of the two must have no more than $k$ crossings. We prove a general upper bound on the number of edges of min-$k$-planar drawings, a finer upper bound for $k=3$, and tight upper bounds for $k=1,2$. Also, we study the inclusion relations between min-$k$-planar graphs (i.e., graphs admitting min-$k$-planar drawings) and $k$-planar graphs. In our setting we only allow simple drawings, that is, any two edges cross at most once, no two adjacent edges cross, and no three edges intersect at a common crossing point., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity. In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model.

Several types of linear layouts of graphs are obtained by leveraging known data structures; the most notable representatives are the stack and the queue layouts. In this content, given a data structure, one seeks to specify an order of the vertices of the graph and a partition of its edges into pages, such that the endpoints of the edges assigned to each page can be processed by the given data structure in the underlying order. In this paper, we study deque and rique layouts of graphs obtained by leveraging the double-ended queue and the restricted-input double-ended queue (or deque and rique, for short), respectively. Hence, they generalize both the stack and the queue layouts. We focus on complete and complete bipartite graphs and present bounds on their deque- and rique-numbers, that is, on the minimum number of pages needed by any of these two types of linear layouts.

A linear layout of a graph $ G $ consists of a linear order $\prec$ of the vertices and a partition of the edges. A part is called a queue (stack) if no two edges nest (cross), that is, two edges $ (v,w) $ and $ (x,y) $ with $ v \prec x \prec y \prec w $ ($ v \prec x \prec w \prec y $) may not be in the same queue (stack). The best known lower and upper bounds for the number of queues needed for planar graphs are 4 [Alam et al., Algorithmica 2020] and 42 [Bekos et al., Algorithmica 2022], respectively. While queue layouts of special classes of planar graphs have received increased attention following the breakthrough result of [Dujmovi\'c et al., J. ACM 2020], the meaningful class of bipartite planar graphs has remained elusive so far, explicitly asked for by Bekos et al. In this paper we investigate bipartite planar graphs and give an improved upper bound of 28 by refining existing techniques. In contrast, we show that two queues or one queue together with one stack do not suffice; the latter answers an open question by Pupyrev [GD 2018]. We further investigate subclasses of bipartite planar graphs and give improved upper bounds; in particular we construct 5-queue layouts for 2-degenerate quadrangulations., Comment: An extended abstract of this paper appears in the Proceedings of the 18th International Symposium on Algorithms and Data Structures, WADS 2023

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 constructing 2-layer drawings often try to minimize the number of edge crossings. We use vertex splitting to reduce the number of crossings, by replacing selected vertices on one layer by two (or more) copies and suitably distributing their incident edges among these copies. We study several optimization problems related to vertex splitting, either minimizing the number of crossings or removing all crossings with fewest splits. While we prove that some variants are \NP-complete, we obtain polynomial-time algorithms for others. We run our algorithms on a benchmark set of bipartite graphs representing the relationships between human anatomical structures and cell types.

This paper studies a \emph{packing} problem in the so-called beyond-planar setting, that is when the host graph is ``almost-planar'' in some sense. Precisely, we consider the case that the host graph is $k$-planar, i.e., it admits an embedding with at most $k$ crossings per edge, and focus on families of $\Delta$-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree $\Delta$. We study the dependency of $k$ from the number $h$ of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called \emph{placement}) and when they are not. We give necessary and sufficient conditions for the placement of $h$ $\Delta$-regular caterpillars and sufficient conditions for the packing of a set of $\Delta_1$-, $\Delta_2$-, $\dots$, $\Delta_h$-regular caterpillars such that the degree $\Delta_i$ and the degree $\Delta_j$ of the non-leaf vertices can differ from one caterpillar to another, for $1 \leq i,j \leq h$, $i\neq j$.

A graph is rectilinear planar if it admits a planar orthogonal drawing without bends. While testing rectilinear planarity is NP-hard in general (Garg and Tamassia, 2001), it is a long-standing open problem to establish a tight upper bound on its complexity for partial 2-trees, i.e., graphs whose biconnected components are series-parallel. We describe a new O(n^2)-time algorithm to test rectilinear planarity of partial 2-trees, which improves over the current best bound of O(n^3 \log n) (Di Giacomo et al., 2022). Moreover, for partial 2-trees where no two parallel-components in a biconnected component share a pole, we are able to achieve optimal O(n)-time complexity. Our algorithms are based on an extensive study and a deeper understanding of the notion of orthogonal spirality, introduced several years ago (Di Battista et al, 1998) to describe how much an orthogonal drawing of a subgraph is rolled-up in an orthogonal drawing of the graph., Comment: arXiv admin note: substantial text overlap with arXiv:2110.00548 Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022)

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 work, we make progress on the problem of finding RAC drawings of graphs of low degree. In this context, a long-standing open question asks whether all degree-3 graphs admit straight-line RAC drawings. This question has been positively answered for the Hamiltonian degree-3 graphs. We improve on this result by extending to the class of 3-edge-colorable degree-3 graphs. When each edge is allowed to have one bend, we prove that degree-4 graphs admit such RAC drawings, a result which was previously known only for degree-3 graphs. Finally, we show that 7-edge-colorable degree-7 graphs admit RAC drawings with two bends per edge. This improves over the previous result on degree-6 graphs., Comment: Extended version of a paper presented at MFCS 2022

A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of $G$ to a sequence of horizontal and vertical segments between its end-points. A longstanding open question in Graph Drawing, dating back over 30 years, is whether there exists a linear-time algorithm to compute an orthogonal drawing of a plane 4-graph with the minimum number of bends. The term "plane" indicates that the input graph comes together with a planar embedding, which must be preserved by the drawing (i.e., the drawing must have the same set of faces as the input graph). In this paper, we positively answer the question above for the widely-studied class of series-parallel graphs. Our linear-time algorithm is based on a characterization of the planar series-parallel graphs that admit an orthogonal drawing without bends. This characterization is given in terms of the orthogonal spirality that each type of triconnected component of the graph can take; the orthogonal spirality of a component measures how much that component is "rolled-up" in an orthogonal drawing of the graph., Comment: arXiv admin note: text overlap with arXiv:2008.03784

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 classes for which the product structure holds, such as to planar graphs [Dujmovi\'c et al., J. ACM, 67(4), 22:1-38, 2020]. In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size $3\lfloor h/2 \rfloor +\lfloor h/3 \rfloor -1$. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 495 to 81 and from 32225k(k-3) to 61k, respectively. We also employ the product structure machinery to improve the current upper bounds of twin-width of planar and 1-planar graphs from 183 to 37, and from O(1) to 80, respectively. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions.

Entity relationship extraction envisions the automatic generation of semantic data models from collections of text, by automatic recognition of entities, by association of entities to form relationships, and by classifying these instances to assign them to entity sets (or classes) and relationship sets (or associations). As a first step in this direction, the Lokahi prototype can extract entities based on the TF*IDF measure, and generate semantic relationships based on document-level co-occurrence statistics, for example with likelihood ratios and pointwise mutual information. This paper presents results of an explorative, prototypical, qualitative and synthetic research, summarizes insights from two research projects and, based on this, indicates an outline for further research in the field of entity relationship extraction from text., Comment: Proceedings of the AAAI 2019 Spring Symposium on Combining Machine Learning with Knowledge Engineering (AAAI-MAKE 2019)

We study the long-standing open problem of efficiently testing rectilinear planarity of series-parallel graphs (SP-graphs) in the variable embedding setting. A key ingredient behind the design of a linear-time testing algorithm for SP-graphs of vertex-degree at most three is that one can restrict the attention to a constant number of ``rectilinear shapes'' for each series or parallel component. To formally describe these shapes the notion of spirality can be used. This key ingredient no longer holds for SP-graphs with vertices of degree four, as we prove a logarithmic lower bound on the spirality of their components. The bound holds even for the independent-parallel SP-graphs, in which no two parallel components share a pole. Nonetheless, by studying the spirality properties of the independent-parallel SP-graphs, we are able to design a linear-time rectilinear planarity testing algorithm for this graph family.

In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs, the recognition problem has been settled, namely it is NP-complete for the general case, while optimal 1-planar graphs, i.e. those with maximum density, can be recognized in linear time. For 2-planar graphs, the picture is less complete. As expected, the recognition problem has been found to be NP-complete in general. In this paper, we consider the recognition of simple optimal 2-planar graphs. We exploit a combinatorial characterization of such graphs and present a linear time algorithm for recognition and embedding., Comment: Appears in the Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021)

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 allowing each set of edges to form either a stack or a queue. In this work we initiate the study of the mixed page number of a graph which corresponds to the minimum number of such sets. First, we study the edge density of graphs with bounded mixed page number. Then, we focus on complete and complete bipartite graphs, for which we derive lower and upper bounds on their mixed page number. Our findings indicate that combining stacks and queues is more powerful in various ways compared to the two traditional layouts.

Following an analysis of the advantages of SQL-based Machine Learning (ML) and a short literature survey of the field, we describe a novel method for In-Database Machine Learning (IDBML). We contribute a process for SQL-code generation in Python using template macros in Jinja2 as well as the prototype implementation of the process. We describe our implementation of the process to compute multidimensional histogram (MDH) probability estimation in SQL. For this, we contribute and implement a novel discretization method called equal quantized rank binning (EQRB) and equal-width binning (EWB). Based on this, we provide data gathered in a benchmarking experiment for the quantitative empirical evaluation of our method and system using the Covertype dataset. We measured accuracy and computation time and compared it to Scikit Learn state of the art classification algorithms. Using EWB, our multidimensional probability estimation was the fastest of all tested algorithms, while being only 1-2% less accurate than the best state of the art methods found (decision trees and random forests). Our method was significantly more accurate than Naive Bayes, which assumes independent one-dimensional probabilities and/or densities. Also, our method was significantly more accurate and faster than logistic regression. This motivates for further research in accuracy improvement and in IDBML with SQL code generation for big data and larger-than-memory datasets.

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 and thoroughly analyze lazy linear extensions for posets of width w > 2. Our analysis implies an upper bound of $(w-1)^2 +1$ on the queue number of width-w posets, which is tight for the strategy and yields an improvement over the previously best-known bound. Further, we provide an example of a poset that requires at least w+1 queues in every linear extension, thereby disproving the conjecture for posets of width w > 2., Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020)

A plane graph is rectilinear planar if it admits an embedding-preserving straight-line drawing where each edge is either horizontal or vertical. We prove that rectilinear planarity testing can be solved in optimal $O(n)$ time for any plane series-parallel graph $G$ with $n$ vertices. If $G$ is rectilinear planar, an embedding-preserving rectilinear planar drawing of $G$ can be constructed in $O(n)$ time. Our result is based on a characterization of rectilinear planar series-parallel graphs in terms of intervals of orthogonal spirality that their components can have, and it leads to an algorithm that can be easily implemented., Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020)

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 graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this problem by demonstrating planar graphs that require four pages in any of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.

We present a prototype online system to automate the procedure of computing different types of linear layouts of graphs under different user-specific constraints. Currently, four different types of linear layouts are supported: stack, queue, rique and deque, as well as, any mixture of them. 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 with several additional features to allow the user to define and further constraint the linear layout to be computed. The server side, which is available to multiple clients through a well-documented API, is responsible for the actual computation of the linear layout. Its algorithmic core is an extension of a SAT formulation that is known to be robust enough to solve non-trivial instances in reasonable amount of time.

We show that every planar graph can be represented by a monotone topological 2-page book embedding where at most 15n/16 (of potentially 3n-6) edges cross the spine exactly once.

In this paper, we study fan-planar drawings that use $h$ layers and are proper, i.e., edges connect adjacent layers. We show that if the embedding of the graph is fixed, then testing the existence of such drawings is fixed-parameter tractable in $h$, via a reduction to a similar result for planar graphs by Dujmovi\'{c} et al. If the embedding is not fixed, then we give partial results for $h=2$: It was already known how to test existence of fan-planar proper 2-layer drawings for 2-connected graphs, and we show here how to test this for trees. Along the way, we exhibit other interesting results for graphs with a fan-planar proper $h$-layer drawings; in particular we bound their pathwidth and show that they have a bar-1-visibility representation.

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 components for two meaningful subclasses of planar graphs, namely maximal outerplanar graphs and complete planar 3-trees.