Walter Didimo

• University of Perugia

A planar orthogonal drawing {\Gamma} of a connected planar graph G is a geometric representation of G such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and no two edges intersect except at common end-points. A bend of {\Gamma} is a point of an edge where a horizontal...

Computing a Euclidean minimum spanning tree of a set of points is a seminal problem in computational geometry and geometric graph theory. We combine it with another classical problem in graph drawing, namely computing a monotone geometric representation of a given graph. More formally, given a finite set $S$ of points in the plane and a finite set...

topological graph (AT-graph) is a pair $A=(G,\mathcal{X})$, where $G=(V,E)$ is a graph and $\mathcal{X} \subseteq {E \choose 2}$ is a set of pairs of edges of $G$. A realization of $A$ is a drawing $\Gamma_A$ of $G$ in the plane such that any two edges $e_1,e_2$ of $G$ cross in $\Gamma_A$ if and only if $(e_1,e_2) \in \mathcal{X}$; $\Gamma_A$ is si...

Computing planar orthogonal drawings with the minimum number of bends is one of the most studied topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia in SIAM J Comput 31(2):601–625, 2001). From the parameteri...

This paper presents a new decision support system offered for an in-depth analysis of semantic networks, which can provide insights for a better exploration of a brand's image and the improvement of its connectivity. In terms of network analysis, we show that this goal is achieved by solving an extended version of the Maximum Betweenness Improvemen...

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...

Computing planar orthogonal drawings with the minimum number of bends is one of the most relevant topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001). From the parameterized complexity perspective, th...

A vertex of a plane digraph is bimodal if all its incoming edges (and hence all its outgoing edges) are consecutive in the cyclic order around it. A plane digraph is bimodal if all its vertices are bimodal. Bimodality is at the heart of many types of graph layouts, such as upward drawings, level-planar drawings, and L-drawings. If the graph is not...

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 \((k \ge 1)\), where each edge cannot cross more than k times. We generalize k-planar drawi...

Graph drawing beyond planarity is a research area that has received an increasing attention in the last twenty years, driven by the necessity to mitigate the visual complexity inherent in geometric representations of non-planar graphs. This research area stems from the study of graph layouts with forbidden crossing configurations, a well-establishe...

The problem of orienting the edges of an undirected graph such that the resulting digraph is acyclic and has a single source $s$ and a single sink $t$ has a long tradition in graph theory and is central to many graph drawing algorithms. Such an orientation is called an $st$-orientation. We address the problem of computing $st$-orientations of undir...

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 descri...

In a graph story the vertices enter a graph one at a time and each vertex persists in the graph for a fixed amount of time $\omega$, called viewing window. At any time, the user can see only the drawing of the graph induced by the vertices in the viewing window and this determines a sequence of drawings. For readability, we require that all the dra...

The study of nonplanar graph drawings with forbidden or desired crossing configurations has a long tradition in geometric graph theory, and received an increasing attention in the last two decades, under the name of beyond-planar graph drawing. In this context, we introduce a new hierarchy of graph families, called \(k^+\)-real face graphs. For any...

A vertex of a plane digraph is bimodal if all its incoming edges (and hence all its outgoing edges) are consecutive in the cyclic order around it. A plane digraph is bimodal if all its vertices are bimodal. Bimodality is at the heart of many types of graph layouts, such as upward drawings, level-planar drawings, and L-drawings. If the graph is not...

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...

Computing planar orthogonal drawings with the minimum number of bends is one of the most relevant topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001). From the parameterized complexity perspective, th...

A k-page upward book embedding (kUBE) of a directed acyclic graph G is a book embeddings of G on k pages with the additional requirement that the vertices appear in a topological ordering along the spine of the book. The k UBE Testing problem, which asks whether a graph admits a kUBE, was introduced in 1999 by Heath, Pemmaraju, and Trenk (SIAM J Co...

This paper addresses the goal of automated creation of intuitive graphical representations of textual descriptions, in particular utilizing the popular storyline visualization paradigm - a diagram that describes a temporal sequence of interactions among several actors. We propose a container-based architecture that integrates natural language proce...

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 polygonal chain consisting of horizontal and vertical segments. A longstanding open question in Graph Drawing, dating back over 30 yea...

Hybrid visualizations combine different metaphors into a single network layout, in order to help humans in finding the “right way” of displaying the different portions of the network, especially when it is globally sparse and locally dense. We investigate hybrid visualizations in two complementary directions: (i) On the one hand, we evaluate the ef...

Orthogonal graph drawings are used in applications such as UML diagrams, VLSI layout, cable plans, and metro maps. We focus on drawing planar graphs and assume that we are given an that describes the desired shape, but not the exact coordinates of a drawing. Our aim is to compute an orthogonal drawing on the grid that has minimum area among all gri...

In a graph story the vertices enter a graph one at a time and each vertex persists in the graph for a fixed amount of time $\omega$, called viewing window. At any time, the user can see only the drawing of the graph induced by the vertices in the viewing window and this determines a sequence of drawings. For readability, we require that all the dra...

A graph is rectilinear planar if it admits a planar orthogonal drawing without bends. While testing rectilinear planarity is NP-hard in general, 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 $O(n^2 \log^2 n)$...

The problem of orienting the edges of an undirected graph such that the resulting digraph is acyclic and has a single source s and a single sink t has a long tradition in graph theory and is central to many graph drawing algorithms. Such an orientation is called an st-orientation. We address the problem of computing st-orientations of undirected gr...

In social networks, individuals’ decisions are strongly influenced by recommendations from their friends, acquaintances, and favorite renowned personalities. The popularity of online social networking platforms makes them the prime venues to advertise products and promote opinions. The Influence Maximization (IM) problem entails selecting a seed...

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 assign...

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...

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...

Many papers study the natural problem of drawing nonplanar graphs with few crossings per edge. In particular, a graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time and several efficient algorithms have been described in the literature, deciding...

We study how to place arrow heads in directed graph drawings aiming at minimizing their overlaps and avoiding intersections between arrow heads and edges. The objective is to support users to correctly and quickly recognize edge orientations, i.e. to deduce unambiguously the edge orientations. Our contribution is two‐fold: (i) We present exact and...

A recent stream of research focuses on building high-performance data analysis and management systems that run completely in the browser. Indeed, today personal devices offer non-trivial amount of computing power, while the latest Web browsers provide powerful JavaScript engines. On the other hand, the use of visualization to present and analyze ne...

Hybrid visualizations mix different metaphors in a single layout of a network. In particular, the popular NodeTrix model, introduced by Henry, Fekete, and McGuffin in 2007, combines node-link diagrams and matrix-based representations to support the analysis of real-world networks that are globally sparse but locally dense. That idea inspired a seri...

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...

Hybrid visualizations mix different metaphors in a single layout of a network. In particular, the popular NodeTrix model, introduced by Henry, Fekete, and McGuffin in 2007, combines node-link diagrams and matrix-based representations to support the analysis of real-world networks that are globally sparse but locally dense. That idea inspired a seri...

This paper describes the research activity on financial crime detection developed by the computer engineering group at the University of Perugia. The presented research aims at designing and experimenting advanced visual interfaces to support financial crime detection, with a focus on tax evasion discovery. The activity of the group on this topic,...

Storyline visualizations depict the temporal dynamics of social interactions, as they describe how groups of actors (individuals or organizations) change over time. A common constraint in storyline visualizations is that an actor cannot belong to two different groups at the same time instant. However, this constraint may be too severe in some appli...

One of the primary goals of many systems for the visual analysis of dynamically changing networks is to maintain the stability of the drawing throughout the sequence of graph changes. We investigate the scenario where the changes are determined by a stream of events, each being either an edge addition or an edge removal. The visualization must be u...

In a RAC drawing of a graph, every two crossing edges form π2 angles at their crossing point. The theoretical study of this type of drawings started in 2009, motivated by cognitive experiments showing that crossings with large angles do not affect too much the readability of a graph layout. Since then, the RAC drawing convention has been widely stu...

In social networks, individuals' decisions are strongly influenced by recommendations from their friends and acquaintances. The influence maximization (IM) problem asks to select a seed set of users that maximizes the influence spread, i.e., the expected number of users influenced through a stochastic diffusion process triggered by the seeds. In th...

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...

Many real-world networks are globally sparse but locally dense. Typical examples are social networks, biological networks, and information networks. This double structural nature makes it difficult to adopt a homogeneous visualization model that clearly conveys both an overview of the network and the internal structure of its communities at the sam...

Storyline visualizations depict the temporal dynamics of social interactions, as they describe how groups of actors (individuals or organizations) change over time. A common constraint in storyline visualizations is that an actor cannot belong to two different groups at the same time instant. However, this constraint may be too severe in some appli...

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...

A graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. The 1-planarity testing problem is NP-complete, even for restricted classes of graphs. We present the first general 1-planarity testing and embedding algorithm, and we experimentally investigate its feasibility in practice. The results suggest that our approach...

This paper presents a novel approach, called MALDIVE, to support tax administrations in the tax risk assessment for discovering tax evasion and tax avoidance. MALDIVE relies on a network model describing several kinds of relationships among taxpayers. Our approach suitably combines various data mining and visual analytics methods to support public...

In social networks, individuals’ decisions are strongly influenced by recommendations from their friends and acquaintances. The influence maximization (IM) problem asks to select a seed set of users that maximizes the influence spread, i.e., the expected number of users influenced through a stochastic diffusion process triggered by the seeds. In th...

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...

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...

Many real-world networks are globally sparse but locally dense. Typical examples are social networks, biological networks, and information networks. This double structural nature makes it difficult to adopt a homogeneous visualization model that clearly conveys an overview of the network and the internal structure of its communities at the same tim...

The definition of $1$-planar graphs naturally extends graph planarity, namely a graph is $1$-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time, deciding whether a graph is $1$-planar is NP-complete, even for restricted classes of graphs. Although severa...

A planar orthogonal drawing $\Gamma$ of a planar graph $G$ is a geometric representation of $G$ such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and no two edges intersect except at their common end-points. A bend of $\Gamma$ is a point of an edge where a horizontal...

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...

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 $...

Many real-world networks are globally sparse but locally dense. Typical examples are social networks, biological networks, and information networks. This double structural nature makes it difficult to adopt a homogeneous visualization model that clearly conveys an overview of the network and the internal structure of its communities at the same tim...

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...

We present TeFNet, a new system for the visual analysis of temporal networks in the fiscal domain, aimed to contrast tax evasion, fiscal frauds, and money laundering. The design of TeFNet has been driven by domain experts (tax officers) and the system is currently used by the Italian revenue agency, Agenzia delle Entrate. TeFNet is based on a power...

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...

We study $k$-page upward book embeddings ($k$UBEs) of $st$-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on $k$ pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a $k$UBE is NP-complete...

The ply number of a drawing is a new criterion of interest for graph drawing. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the longest edge incident to that vertex. This paper reports the...

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...

We prove that every set \(\mathcal {S}\) of \(\varDelta \) slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree \(\varDelta \), 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-...

Let G be a planar 3-graph (i.e., a planar graph with vertex degree at most three) with n vertices. We present the first \(O(n^2)\)-time algorithm that computes a planar orthogonal drawing of G with the minimum number of bends in the variable embedding setting. If either a distinguished edge or a distinguished vertex of G is constrained to be on the...

The use of graph visualization approaches to present and analyze complex data is taking a leading role in conveying information and knowledge to users in many application domains. This creates the need of developing efficient and effective algorithms that automatically compute graph layouts. In this respect, force-directed algorithms are arguably a...

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 3-p...

An HV-graph is a planar graph with vertex-degree at most four such that each edge is labeled either H (horizontal) or V (vertical). The HV-planarity testing problem asks whether an HV-graph admits an HV-drawing, that is, a planar drawing such that each edge with label H is drawn as a horizontal segment and each edge with label V is drawn as a verti...

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...

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...

Analyzing large-scale graphs provides valuable insights in different application scenarios, including social networking, crime detection, content ranking, and recommendations. While many graph processing systems working on top of distributed infrastructures have been proposed to deal with big graphs, the task of profiling their massive computations...

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of non-planar graphs in terms of forbidden crossing configurations. Aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of non-planar graphs in terms of forbidden crossing configurations. Aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

Let $G$ be a planar $3$-graph (i.e., a planar graph with vertex degree at most three) with $n$ vertices. We prove that a planar orthogonal drawing of $G$ with the minimum number of bends can be computed in $O(n^2)$ time. The minimum is taken over all planar embeddings of $G$. The most efficient known algorithm to solve this problem has complexity $...

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...

This paper describes TAXNET, a decision support system for tax evasion discovery, based on a powerful visual language and on advanced network visualization techniques. It has been developed in cooperation with the Italian Revenue Agency, where it is currently used. TAXNET allows users to visually define classes of suspicious patterns, it exploits e...

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...

A graph is k-planar \((k \ge 1)\) if it can be drawn in the plane such that no edge is crossed \(k+1\) times or more. A graph is k-quasi-planar \((k \ge 2)\) if it can be drawn in the plane with no k pairwise crossing edges. The families of k-planar and k-quasi-planar graphs have been widely studied in the literature, and several bounds have been p...

Overloaded orthogonal drawing (OOD) is a recent graph visualization style specifically conceived for directed graphs. It merges the advantages of some popular drawing conventions like layered drawings and orthogonal drawings, and provides additional support for some common analysis tasks. We present a visualization framework called DAGView, which i...

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with...

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with...

Analyzing large-scale graphs provides valuable insights in different application scenarios. While many graph processing systems working on top of distributed infrastructures have been proposed to deal with big graphs, the tasks of profiling and debugging their massive computations remain time consuming and error-prone. This paper presents GiViP, a...

A $1$-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a $1$-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a $1$-plane graph $G$ is a coloring of the edges of $G$ with two colors, red and blue, such that both the graph induced by...

A $1$-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a $1$-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a $1$-plane graph $G$ is a coloring of the edges of $G$ with two colors, red and blue, such that both the graph induced by...

A graph is $k$-planar $(k \geq 1)$ if it can be drawn in the plane such that no edge is crossed more than $k$ times. A graph is $k$-quasi planar $(k \geq 2)$ if it can be drawn in the plane with no $k$ pairwise crossing edges. The families of $k$-planar and $k$-quasi planar graphs have been widely studied in the literature, and several bounds have...

The ply number of a drawing is a new criterion of interest for graph drawing. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the longest edge incident to that vertex. This paper reports the...

Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with...

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at mo...

We consider the problem of placing arrow heads in directed graph drawings without them overlapping other drawn objects. This gives drawings where edge directions can be deduced unambiguously. We show hardness of the problem, present exact and heuristic algorithms, and report on a practical study.

Big Data analytics is recognized as one of the major issues in our current information society, and raises several challenges and opportunities in many fields, including economy and finance, e-commerce, public health and administration, national security, and scientific research. The use of visualization techniques to make sense of large volumes of...

The wide availability of powerful and inexpensive cloud computing services naturally motivates the study of distributed graph layout algorithms, able to scale to very large graphs. Nowadays, to process Big Data, companies are increasingly relying on PaaS infrastructures rather than buying and maintaining complex and expensive hardware. So far, only...

A graph is 1-planar if it has a drawing where each edge is crossed at most once. A drawing is RAC (Right Angle Crossing) if the edges cross only at right angles. The relationships between 1-planar graphs and RAC drawings have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC dra...

We study the problem of computing drawings of planar graphs in sub-quadratic area, by allowing edge crossings. We first prove that sub-quadratic area cannot be achieved if only a constant number of crossings per edge is allowed. More precisely, we show that the same area lower bounds as in the crossing-free case hold for straight-line and poly-line...

A graph is 1-planar if it has a drawing where each edge is crossed at most once. A drawing is RAC (Right Angle Crossing) if the edges cross only at right angles. The relationships between 1-planar graphs and RAC drawings have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC dra...