Unit edge-length drawings, rectilinear drawings (where each edge is either a horizontal or a vertical segment), and rectangular face drawings are among the most studied subjects in Graph Drawing. However, most of the literature on these topics refers to planar graphs and planar drawings. In this paper we study drawings with all the above nice properties but that can have edge crossings; we call them Unit Edge length Rectilinear drawings with Rectangular Faces (UER-RF drawings). We consider crossings as dummy vertices and apply the unit edge-length convention to the edge segments connecting any two (real or dummy) vertices. Note that UER-RF drawings are grid drawings (vertices are placed at distinct integer coordinates), which is another classical requirement of graph visualizations. We present several efficient and easily implementable algorithms for recognizing graphs that admit UER-RF drawings and for constructing such drawings if they exist. We consider restrictions on the degree of the vertices or on the size of the faces. For each type of restriction, we consider both the general unconstrained setting and a setting in which either the external boundary of the drawing is fixed or the rotation system of the graph is fixed as part of the input.

We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set S of points, while ensuring that its curve complexity (i.e., maximum number of bends per edge) is bounded by a constant. We focus on trees: Let T be a tree, ϑ(T) be its thrackle number, and χ be any integer in the interval [0,ϑ(T)]. In the tangling phase we compute a topological linear embedding of T with ϑ(T) edge crossings and a constant number of spine traversals. In the untangling phase we remove edge crossings without increasing the spine traversals until we reach χ crossings. The computed linear embedding is used to construct a drawing of T on S with χ crossings and constant curve complexity. Our approach gives rise to an O(n^2)-time algorithm for general trees and an O(n log n)-time algorithm for paths. We also adapt the approach to compute RAC drawings, i.e. drawings where the angles formed at edge crossings are 2.

A weak dominance drawing Γ of a DAG G=(V,E) is a d-dimensional drawing such that D(u)

Inquiries such as whether a task A depends on a task B, whether an author A has been influenced by a paper B, whether a certain protein is associated with a specific biological process or molecular function, or whether class A inherits from class B, are just a few examples of inquiries that can be modeled as reachability queries on a network (Directed Graph). Digital systems answer myriad such inquiries every day. In this paper, we discuss the transitive closure problem. We focus on applicable solutions that enable us to answer queries fast, in constant time, and can serve in real-world applications. In contrast to the majority of research on this topic that revolves around the construction of a two-dimensional adjacency matrix, we present an approach that builds a reachability indexing scheme. This scheme enables us to answer queries in constant time and can be built in parameterized linear time. In addition, it captures a compressed data structure. Our approach and algorithms are validated by extensive experiments that shed light on the factors that play a key role in this problem. To stress the efficiency of this solution and demonstrate the potential to apply our approach to important problems, we use it algorithm to speed up Fulkerson's method for finding the width of a DAG. Our results challenge the prevailing belief, reiterated over the last thirty years, regarding the efficiency of this method. Our approach is based on the concept of chain decomposition. Before we delve into its description, we introduce, analyze, and utilize a chain decomposition algorithm. Furthermore, we explore how chain decomposition can facilitate transitive closure solutions introducing a general purpose linear time reduction technique that removes a large subset of transitive edges given any chain decomposition.

We present a detailed hierarchical graph drawing technique that is based on the Path Based Framework (PBF) . Extensive edge bundling is applied to draw all edges of the graph and the height of the drawing is minimized using compaction. We present experimental results and a user study for hierarchical drawings of graphs that show the usefulness of the new drawing technique. The drawings produced by the new framework are compared to drawings produced by the well known Sugiyama framework in terms of area, number of bends, number of crossings, and execution time. The new algorithm runs very fast and produces drawings that are readable and more efficient in terms of area and number of bends. Furthermore, since there are advantages (and disadvantages) to both frameworks, we performed a user study. The results show that the drawings produced by the new framework are well received in terms of clarity, readability, and usability. Hence, the new technique offers an interesting alternative to drawing hierarchical graphs, and is especially useful in applications where user defined paths are important and need to be highlighted.

We present a fast and practical algorithm to compute the transitive closure (TC) of a directed graph. It is based on computing a reachability indexing scheme of a directed acyclic graph (DAG), G = ( V, E ). Given any path/chain decomposition of G we show how to compute in parameterized linear time such a reachability scheme that can answer reachability queries in constant time. The experimental results reveal that our method is significantly faster in practice than the theoretical bounds imply, indicating that path/chain decomposition algorithms can be applied to obtain fast and practical solutions to the transitive closure (TC) problem. Furthermore, we show that the number of non-transitive edges of a DAG G is ≤ width ∗ | V | and that we can find a substantially large subset of the transitive edges of G in linear time using a path/chain decomposition. Our extensive experimental results show the interplay between these concepts in various models of DAGs. 2012 ACM Subject Classification Theory of computation → Theory and algorithms for application domains; Theory of computation → Design and analysis of algorithms

A weak dominance drawing $$\varGamma $$ of a DAG $$G=(V,E)$$ is a d-dimensional drawing such that $$D(u) <D(v)$$ for every dimension D of $$\varGamma $$ if there is a directed path from a vertex u to a vertex v in G, where D(w) is the coordinate of vertex $$w\in V$$ in dimension D of $$\varGamma $$ . If $$D(u) < D(v)$$ for every dimension D of  $$\varGamma $$ , but there is no path from u to v, we have a falsely implied path (fip). Minimizing the number of fips is an important theoretical and practical problem, which is NP-hard. We show that it is an FPT problem for graphs having bounded modular width mw and when d is bounded. This result in weak dominance, which is interesting by itself, lets us prove our main contributions. Computing the dominance dimension of G, that is, the minimum number of dimensions for which G has a dominance drawing (a weak dominance drawing with 0 fips), is a well-known NP-hard problem. We show that the dominance dimension of G is bounded by $$\frac{mw}{2}$$ (mw, if $$mw<4$$ ) and that computing the dominance dimension of G is an FPT problem with parameter mw.

We present a new heuristic algorithm for computing a minimum Feedback Arc Set in directed graphs. The new technique produces solutions that are better than the ones produced by the best previously known heuristics, often reducing the FAS size by more than 50%. It is based on computing the PageRank score of the nodes of the directed line graph of the input directed graph. Although the time required by our heuristic is heavily influenced by the size of the produced line graph, our experimental results show that it runs very fast even for very large graphs used in graph drawing.

We present practical linear and almost linear-time algorithms to compute a chain decomposition of a directed acyclic graph (DAG), $G=(V,E)$. The number of vertex-disjoint chains computed is very close to the minimum. The time complexity of our algorithm is $O(|E|+c*l)$, where $c$ is the number of path concatenations and $l$ is the length of a longest path of the graph. We give a comprehensive explanation on factors $c$ and $l$ in the following sections. Our techniques have important applications in many areas, including the design of faster practical transitive closure algorithms. We observe that $|E_{red}|\leq width*|V|$ ($E_{red}$: non-transitive edges) and show how to find a substantially large subset of $E_{tr}$ (transitive edges) using a chain decomposition in linear time, without calculating the transitive closure. Our extensive experimental results show the interplay between the width, $E_{red}$, $E_{tr}$ in various models of graphs. We show how to compute a reachability indexing scheme in $O(k_c*|E_{red}|)$ time, where $k_c$ is the number of chains and $|E_{red}|$ is the number of non-transitive edges. This scheme can answer reachabilitiy queries in constant time. The space complexity of the scheme is $O(k_c*|V|)$. The experimental results reveal that our methods are even better in practice than the theoretical bounds imply, indicating how fast chain decomposition algorithms can be applied to the transitive closure problem.

A weak dominance drawing Γ of a DAG G=(V,E), is a d-dimensional drawing such that there is a directed path from a vertex u to a vertex v in G if D(u) <D(v) for every dimension D of Γ. We have a falsely implied path (fip) when D(u) < D(v) for every dimension D of Γ, but there is no path from u to v. Minimizing the number of fips is an important theoretical and practical problem, which is NP-hard. We show that it is an FPT problem for parameter k, where k is the maximum degree of a vertex of the modular decomposition tree of G. Namely, for any constant d, we present an O(nm+ndk^2(k!)^d) time algorithm to compute a weak d-dimensional dominance drawing Γ of a DAG G having the minimum number of fips. An interesting implication of this result is that we can decide if a DAG has dominance dimension 3 (a well-known NP-complete problem) in time O(nm+nk^2(k!)^3).

In a dominance drawing Γ of a directed acyclic graph (DAG) G, a vertex v is reachable from a vertex u if, and only if all the coordinates of v are greater than or equal to the coordinates of u in Γ. Dominance drawings of DAGs are very important in many areas of research. They combine the aspect of drawing a DAG on the grid with the fact that the transitive closure of the DAG is apparently obvious by the dominance relation between grid points associated with the vertices. The smallest number d for which a given DAG G has a d-dimensional dominance drawing is called dominance drawing dimension, and it is NP-hard to compute. In this paper, we present efficient algorithms for computing dominance drawings of G with a number of dimensions respecting theoretical bounds. We first describe a simple algorithm that shows how to compute a dominance drawing of G from its compressed transitive closure. Next, we describe a more complicated algorithm, which is based on the concept of modular decomposition of G, and obtaining dominance drawings with a lower number of dimensions. Finally, we consider the concept of weak dominance, a relaxed version of the dominance, and we discuss interesting experimental results.

Answering reachability queries in directed acyclic graphs is an operation required by many applications. In this paper, we present efficient algorithms to construct and search a space-efficient data structure in the k -dimensional space that is based on Graph Dominance Drawing. Our algorithms construct this data structure in O ( km ) time, while it can be stored in O ( kn ) space. Any reachability query is answered in constant time, since no “falsely implied paths (fips)” are introduced. We also present experimental results that show that the number of dimensions, k , in the solutions produced by our techniques is low. Additionally, we present a new method for constructing random DAGs with prespecified structure and density. The analysis of our experimental results reveals an interesting interplay between density and structure.

We present algorithms that extend the path-based hierarchical drawing framework and give experimental results. Our algorithms run in $O(km)$ time, where $k$ is the number of paths and $m$ is the number of edges of the graph, and provide better upper bounds than the original path based framework: e.g., the height of the resulting drawings is equal to the length of the longest path of $G$, instead of $n-1$, where $n$ is the number of nodes. Additionally, we extend this framework, by bundling and drawing all the edges of the DAG in $O(m + n \log n)$ time, using minimum extra width per path. We also provide some comparison to a well known hierarchical drawing framework, widely known as the Sugiyama framework, as a proof of concept. The experimental results show that our algorithms produce drawings that are better in area and number of bends, but worse for crossings in sparse graphs. Hence, our technique offers an interesting alternative for drawing hierarchical graphs. Finally, we present an $O(m + k \log k)$ time algorithm that computes a specific order of the paths in order to reduce the total edge length and number of crossings and bends.

We present algorithms and experiments for the visualization of directed graphs that focus on displaying their reachability information. Our algorithms are based on the concepts of the path and channel decomposition as proposed in the framework presented in GD 2018 (pp. 579-592) and focus on showing the existence of paths clearly. In this paper we customize these concepts and present experimental results that clearly show the interplay between bends, crossings and clarity. Additionally, our algorithms have direct applications to the important problem of showing and storing transitivity information of very large graphs and databases. Only a subset of the edges is drawn, thus reducing the visual complexity of the resulting drawing, and the memory requirements for storing the transitivity information. Our algorithms require almost linear time, $O(kn+m)$, where $k$ is the number of paths/channels, $n$ and $m$ is the number of vertices and edges, respectively. They produce progressively more abstract drawings of the input graph. No dummy vertices are introduced and the vertices of each path/channel are vertically aligned.

Let $G$ be a DAG with $n$ vertices and $m$ edges. Two vertices $u,v$ are incomparable if $u$ doesn't reach $v$ and vice versa. We denote by \emph{width} of a DAG $G$, $w_G$, the maximum size of a set of incomparable vertices of $G$. In this paper we present an algorithm that computes a dominance drawing of a DAG G in $k$ dimensions, where $w_G \le k \le \frac{n}{2}$. The time required by the algorithm is $O(kn)$, with a precomputation time of $O(km)$, needed to compute a \emph{compressed transitive closure} of $G$, and extra $O(n^2w_G)$ or $O(n^3)$ time, if we want $k=w_G$. Our algorithm gives a tighter bound to the dominance dimension of a DAG. As corollaries, a new family of graphs having a 2-dimensional dominance drawing and a new upper bound to the dimension of a partial order are obtained. We also introduce the concept of transitive module and dimensional neck, $w_N$, of a DAG $G$ and we show how to improve the results given previously using these concepts.

We present a new approach to visualize directed graphs and their hierarchies that departs from the classical four-phase framework of Sugiyama and computes readable hierarchical visualizations that focus on the reachability information of a directed acyclic graph. Additionally, our approach has the feature that a few transitive edges are not drawn in the drawing, thus reducing the visual complexity of the resulting drawing. Furthermore, the problems involved in our framework require only polynomial time. The channel decomposition is a partition of the vertex set of the graph into channels, where a channel is a relaxed path. Our framework offers a suite of solutions depending upon the requirements, and it consists of only two steps: (a) the cycle removal step (if the directed graph contains cycles) and (b) the channel decomposition and hierarchical drawing step. Our framework does not introduce any dummy vertices and it keeps the vertices of a path/channel vertically aligned. The time complexity of the main drawing algorithms of our framework is $O(km)$, where $k$ is the number of paths/channels, typically much smaller than $n$ (the number of vertices).

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 at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of layered 1-bend drawings.

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 implements algorithms and graphical features for the OOD style. Besides the algorithm for acyclic digraphs, the DAGView framework implements extensions to visualize both digraphs with cycles and undirected graphs, with the additional possibility of taking into account user preferences and constraints. It also supports an interactive visualization of clustered digraphs, based on the use of strongly connected components. Moreover, we describe an experimental user study, aimed to investigate the usability of OOD within the DAGView framework. The results of our study suggest that OOD can be effectively exploited to perform some basic tasks of analysis in a faster and more accurate way when compared to other drawing styles for directed graphs.

In this paper we present a new approach to visualize directed graphs and their hierarchies that completely departs from the classical four-phase framework of Sugiyama and computes readable hierarchical visualizations that contain the complete reachability information of a graph. Additionally, our approach has the advantage that only the necessary edges are drawn in the drawing, thus reducing the visual complexity of the resulting drawing. Furthermore, most problems involved in our framework require only polynomial time. Our framework offers a suite of solutions depending upon the requirements, and it consists of only two steps: (a) the cycle removal step (if the graph contains cycles) and (b) the channel decomposition and hierarchical drawing step. Our framework does not introduce any dummy vertices and it keeps the vertices of a channel vertically aligned. The time complexity of the main drawing algorithms of our framework is O(kn), where k is the number of channels, typically much smaller than n (the number of vertices).

We study planar drawings of directed graphs in the L-drawing standard. We provide necessary conditions for the existence of these drawings and show that testing for the existence of a planar L-drawing is an NP-complete problem. Motivated by this result, we focus on upward-planar L-drawings. We show that directed st-graphs admitting an upward- (resp. upward-rightward-) planar L-drawing are exactly those admitting a bitonic (resp. monotonically increasing) st-ordering. We give a linear-time algorithm that computes a bitonic (resp. monotonically increasing) st-ordering of a planar st-graph or reports that there exists none.