Abstract We introduce C lusterix , a novel hybrid visualization model for representing hierarchically clustered networks, which also supports directed and weighted edges. C lusterix offers an integrated view of both the network and its full cluster hierarchy by compactly visualizing the cluster inclusion tree enriched with links of the network. This is achieved through matrix‐based representations at various hierarchy levels, combined with a node‐link style linear layout at the leaf level. To support layout computation based on C lusterix , we propose two algorithmic approaches: an exact Integer Linear Program and a fast heuristic, both aimed at minimizing edge crossings. We present an extensive experimental comparison of these algorithmic approaches to highlight the trade‐offs between efficiency and effectiveness. Moreover, as a proof of concept for our model, we developed an interactive visualization system based on C lusterix and evaluated its performance through case studies and qualitative feedback from experts in different application domains.

A rectilinear-upward planar drawing of a digraph G is a crossing-free drawing of G where each edge is either a horizontal or a vertical segment, and such that no directed edge points downward. Rectilinear-Upward Planarity Testing is the problem of deciding whether a digraph G admits a rectilinear-upward planar drawing. We study the complexity of Rectilinear-Upward Planarity Testing and provide several algorithmic results. Precisely, we prove that: ( i ) the problem is NP-complete, even if G is biconnected; ( i i ) it can be solved in linear time when an upward planar embedding of G is fixed; ( i i i ) the problem is polynomial-time solvable for biconnected digraphs of treewidth at most two, i.e., for digraphs whose underlying undirected graph is a series-parallel graph; ( i v ) the problem is fixed-parameter tractable (namely, fixed-parameter linear) for all biconnected graphs, when parameterized by the number of sources and sinks in the digraph. • We study the algorithmic complexity of a problem that combines two well-established topics in graph drawing, namely rectilinear planar drawings and upward planar drawings. This problems, called rectilinear-upward planarity testing, asks to decide whether an input planar di-graph admits a planar drawing where each edge is either a horizontal or a vertical segment, and no edge points downwards. • We prove that rectilinear-upward planarity testing is NP-complete, even for biconnected digraphs. • We provide a linear-time algorithm for rectilinear-upward planarity testing of digraphs with a fixed upward planar embedding. • We provide a quadratic-time algorithm for rectilinear-upward planarity testing of biconnected partial 2-trees (i.e., digraphs whose underlying undirected graph is series-parallel) in the variable embedding setting. • We provide a fixed-parameter linear (FPL) algorithm for rectilinear- upward planarity testing of general biconnected digraphs in the variable embedding setting.

Our work contributes to the fast-growing literature on the use of Large Language Models (LLMs) to perform graph-related tasks. In particular, we focus on usage scenarios that rely on the visual modality, feeding the model with a drawing of the graph under analysis. We investigate how the model's performance is affected by the chosen layout paradigm, the aesthetics of the drawing, and the prompting technique used for the queries. We formulate three corresponding research questions and present the results of a thorough experimental analysis. Our findings reveal that choosing the right layout paradigm and optimizing the readability of the input drawing from a human perspective can significantly improve the performance of the model on the given task. Moreover, selecting the most effective prompting technique is a challenging yet crucial task for achieving optimal performance.

A linear layout of a graph defines a total order of the vertices and partitions the edges into either stacks or queues, i.e., crossing-free and non-nested sets of edges along the order, respectively. In this work, we study defective linear layouts that allow forbidden patterns among edges of the same set. Our focus is on k-defective stack layouts and k-defective queue layouts, in which the conflict graph representing the forbidden patterns among the edges of each stack or queue has maximum degree at most k.

We present TReView, the first visual analytics system for the exploration of the European Union (EU) Transparency Register, a large repository that aims to enhance transparency around lobbying activities within the EU, by enabling public oversight of meetings between lobbyists and EU officials.

We address the problem of computing a dynamic visualization of a geometric graph G as a sequence of frames. Each frame shows only a portion of the graph but their union covers G entirely. The two main requirements of our dynamic visualization are: (i) guaranteeing drawing stability, so to preserve the user’s mental map; (ii) keeping the visual complexity of each frame low. To satisfy the first requirement, we never change the position of the vertices. Regarding the second requirement, we avoid edge crossings in each frame. More precisely, in the first frame we visualize a suitable subset of non-crossing edges; in each subsequent frame, exactly one new edge enters the visualization and all the edges that cross with it are deleted. We call such a sequence of frames a planar story of G. Our goal is to find a planar story whose minimum number of edges contemporarily displayed is maximized (i.e., a planar story that maximizes the minimum frame size). Besides studying our model from a theoretical point of view, we also design and experimentally compare different algorithms, both exact techniques and heuristics. These algorithms provide an array of alternative trade-offs between efficiency and effectiveness, also depending on the structure of the input graph.

A linear layout of a graph consists of a linear ordering of its vertices and a partition of its edges into pages such that the edges assigned to the same page obey some constraint. The two most prominent and widely studied types of linear layouts are stack and queue layouts, in which any two edges assigned to the same page are forbidden to cross and nest, respectively. The names of these two layouts derive from the fact that, when parsing the graph according to the linear vertex ordering, the edges in a single page can be stored using a single stack or queue, respectively. Recently, the concepts of stack and queue layouts have been extended by using a double-ended queue or a restricted-input queue for storing the edges of a page. We extend this line of study to edge-weighted graphs by introducing priority queue layouts, that is, the edges on each page are stored in a priority queue whose keys are the edge weights. First, we show that there are edge-weighted graphs that require a linear number of priority queues. Second, we characterize the graphs that admit a priority queue layout with a single queue, regardless of the edge-weight function, and we provide an efficient recognition algorithm. Third, we show that the number of priority queues required independently of the edge-weight function is bounded by the pathwidth of the graph, but can be arbitrarily large already for graphs of treewidth two. Finally, we prove that determining the minimum number of priority queues is NP-complete if the linear ordering of the vertices is fixed.

The edge-length ratio of a planar straight-line drawing Γ of a graph G is the largest ratio between the lengths of every pair of edges of Γ. If the ratio is measured by considering only pairs of edges that are incident to a common vertex, we talk about local edge-length ratio. The (local) edge-length ratio of a planar graph is the infimum over all (local) edge-length ratios of its planar straight-line drawings. It is known that the edge-length ratio of outerplanar graphs is upper bounded by a constant, while there exist graph families with non-constant outerplanarity that have non-constant lower bounds on their edge-length ratios. In this paper we prove an Ω ( n ) lower bound on the local edge-length ratio (and hence on the edge-length ratio) of the n -vertex 2-outerplanar graphs. We also prove a constant upper bound on the edge-length ratio of Halin graphs, pseudo-Halin graphs, and their generalizations.

A drawing of a graph divides the plane into topologically connected regions, called faces (or cells ). The boundary of each face is formed by vertices, crossings, and edge portions. Given a positive integer , we say that is a -real face drawing of if the boundary of each face of contains at least vertices of . Graphs that admit a -real face drawing are -real face graphs ; they have been studied so far in terms of edge density and inclusion relationships with other notable classes of nonplanar graphs that can be drawn avoiding specific crossing configurations. In this paper, we investigate the complexity of recognizing -real face graphs, that is, the complexity of testing whether a given graph is -real face, for desired values of . We study both the general unconstrained scenario and the 2-layer scenario in which the graph is bipartite, the vertices of the two partition sets lie on two distinct horizontal layers, and the edges are drawn as straight-line segments. While we prove NP-completeness results for the unconstrained scenario, we describe efficient recognition algorithms for the 2-layer setting.

An abstract 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 $Γ_A$ of $G$ in the plane such that any two edges $e_1,e_2$ of $G$ cross in $Γ_A$ if and only if $(e_1,e_2) \in \mathcal{X}$; $Γ_A$ is simple if any two edges intersect at most once (either at a common endpoint or at a proper crossing). The AT-graph Realizability (ATR) problem asks whether an input AT-graph admits a realization. The version of this problem that requires a simple realization is called Simple AT-graph Realizability (SATR). It is a classical result that both ATR and SATR are NP-complete. In this paper, we study the SATR problem from a new structural perspective. More precisely, we consider the size $\mathrmλ(A)$ of the largest connected component of the crossing graph of any realization of $A$, i.e., the graph ${\cal C}(A) = (E, \mathcal{X})$. This parameter represents a natural way to measure the level of interplay among edge crossings. First, we prove that SATR is NP-complete when $\mathrmλ(A) \geq 6$. On the positive side, we give an optimal linear-time algorithm that solves SATR when $\mathrmλ(A) \leq 3$ and returns a simple realization if one exists. Our algorithm is based on several ingredients, in particular the reduction to a new embedding problem subject to constraints that require certain pairs of edges to alternate (in the rotation system), and a sequence of transformations that exploit the interplay between alternation constraints and the SPQR-tree and PQ-tree data structures to eventually arrive at a simpler embedding problem that can be solved with standard techniques.

Abstract 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 parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number b of bends, the number k of vertices of degree at most two, and the treewidth $$\textsf{tw}$$ tw of the input graph (Di Giacomo et al. in J Comput Syst Sci 125:129–148, 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by $$b+k$$ b + k . As a consequence, rectilinear planarity testing lies in FPT parameterized by the number of vertices of degree at most two. We also prove that our choice of parameters is minimal, as deciding if an orthogonal drawing with at most b bends exists is already NP-hard when k is zero (i.e., the problem is para-NP-hard parameterized in k); hence, there is neither an FPT nor an XP algorithm parameterized only by the parameter k (unless P = NP). In addition, we prove that the problem is W[1]-hard parameterized by $$k+\textsf{tw}$$ k + tw , complementing a recent result (Jansen et al. in Upward and orthogonal planarity are W[1]-hard parameterized by treewidth. CoRR, abs/2309.01264, 2023; in: Bekos MA, Chimani M (eds) Graph Drawing and Network Visualization, vol 14466, Springer, Cham, pp 203–217, 2023) that shows W[1]-hardness for the parameterization $$b+\textsf{tw}$$ b + tw . As a consequence, we are able to trace a clear parameterized tractability landscape for the bend-minimum orthogonal planarity problem with respect to the three parameters b, k, and $$\textsf{tw}$$ tw .

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 effectiveness of different hybrid visualization models through a comparative user study; (ii) On the other hand, we estimate the usefulness of an interactive visualization that integrates all the considered hybrid models together. The results of our study provide some hints about the usefulness of the different hybrid visualizations for specific tasks of analysis and indicates that integrating different hybrid models into a single visualization may offer a valuable tool of analysis.

Abstract 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 kUBE Testing problem, which asks whether a graph admits a kUBE, was introduced in 1999 by Heath, Pemmaraju, and Trenk (SIAM J Comput 28(4), 1999). In a companion paper, Heath and Pemmaraju (SIAM J Comput 28(5), 1999) proved that the problem is linear-time solvable for $$k=1$$ k = 1 and NP-complete for $$k = 6$$ k = 6 . Closing this gap has been a central question in algorithmic graph theory since then. In this paper, we make a major contribution towards a definitive answer to the above question by showing that kUBE Testing is NP-complete for $$k\ge 3$$ k ≥ 3 , even for st-graphs, i.e., acyclic directed graphs with a single source and a single sink. Indeed, our result, together with a recent work of Bekos et al. (Theor Comput Sci 946, 2023) that proves the NP-completeness of 2UBE for planar st-graphs, closes the question about the complexity of the kUBE problem for any k. Motivated by this hardness result, we then focus on the 2UBE Testing for planar st-graphs. On the algorithmic side, we present an $$O(f(\beta )\cdot n+n^3)$$ O ( f ( β ) · n + n 3 ) -time algorithm for 2UBE Testing, where $$\beta $$ β is the branchwidth of the input graph and f is a singly-exponential function on $$\beta $$ β . Since the treewidth and the branchwidth of a graph are within a constant factor from each other, this result immediately yields an FPT algorithm for st-graphs of bounded treewidth. Furthermore, we describe an O(n)-time algorithm to test whether a plane st-graph whose faces have a special structure admits a 2UBE that additionally preserves the plane embedding of the input st-graph. On the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving $$2$$ 2 UBE.

Abstract 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 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.

Abstract We study universal sets of slopes for computing upward planar drawings of planar st-graphs. We first consider a subfamily of planar st-graphs, called bitonic st-graphs. We prove that every set $$\mathcal {S}$$ S of $$\varDelta $$ Δ slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree $$\varDelta $$ Δ , i.e., every such digraph admits a 1-bend upward planar drawing whose edge segments use only slopes in $$\mathcal {S}$$ S . This result is worst-case optimal in terms of number of slopes, and, for a suitable choice of $$\mathcal {S}$$ S , it gives rise to drawings with worst-case optimal angular resolution. We then prove that every such set $$\mathcal {S}$$ S can be used to construct 2-bend upward planar drawings of n-vertex planar st-graphs with at most $$4n-9$$ 4 n - 9 bends in total.

Abstract 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 heuristic algorithms for this arrow placement problem, along with an extensive experimental analysis of these techniques; and (ii) we report on a user study aimed to understand the impact of different arrow placement strategies on performing global and local analysis tasks on directed graph layouts.

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 same time. As a consequence, the use of hybrid visualizations has been proposed. For instance, NodeTrix combines node-link and matrix-based representations (Henry et al., 2007). In this article we describe ChordLink, a hybrid visualization model that embeds chord diagrams, used to represent dense subgraphs, into a node-link diagram, which shows the global network structure. The visualization makes it possible to interactively highlight the structure of a community while keeping the rest of the layout stable. We discuss the intriguing algorithmic challenges behind the ChordLink model, present a prototype system that implements it, and illustrate case studies on real-world networks.

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 set of users that maximizes the influence spread, i.e., the expected number of users positively influenced by a stochastic diffusion process triggered by the seeds. Engineering and analyzing IM algorithms remains a difficult and demanding task due to the NP-hardness of the problem and the stochastic nature of the diffusion processes. Despite several heuristics being introduced, they often fail in providing enough information on how the network topology affects the diffusion process, precious insights that could help researchers improve their seed set selection. In this paper, we present VAIM, a visual analytics system that supports users in analyzing, evaluating, and comparing information diffusion processes determined by different IM algorithms. Furthermore, VAIM provides useful insights that the analyst can use to modify the seed set of an IM algorithm, so to improve its influence spread. We assess our system by: (i) a qualitative evaluation based on a guided experiment with two domain experts on two different data sets; (ii) a quantitative estimation of the value of the proposed visualization through the ICE-T methodology by Wall et al. (IEEE TVCG - 2018). The twofold assessment indicates that VAIM effectively supports our target users in the visual analysis of the performance of IM algorithms.

This paper addresses a long standing, widely studied, open question: Given a planar 3-graph G (i.e., a planar graph with vertex degree at most three), what is the best computational upper bound to compute a bend-minimum planar orthogonal drawing of G in the variable embedding setting? In this setting the algorithm can choose among the exponentially many planar embeddings of G the one that leads to an orthogonal drawing with the minimum number of bends. We answer the question by describing a linear-time algorithm that computes a bend-minimum planar orthogonal drawing of G. Also, if G is not K4, the drawing has at most one bend per edge. The existence of an orthogonal drawing Г of a planar 3-graph such that Г has the minimum number of bends and at most one bend per edge was previously unknown.

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 for $k\geq 3$. A hardness result for this problem was previously known only for $k = 6$ [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on $k=2$. On the algorithmic side, we present polynomial-time algorithms for testing the existence of $2$UBEs of planar $st$-graphs with branchwidth $β$ and of plane $st$-graphs whose faces have a special structure. These algorithms run in $O(f(β)\cdot n+n^3)$ time and $O(n)$ time, respectively, where $f$ is a singly-exponential function on $β$. Moreover, on the combinatorial side, we present two notable families of plane $st$-graphs that always admit an embedding-preserving $2$UBE.

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 among the most popular graph layout techniques. Aimed at leveraging the potential of modern distributed graph algorithms platforms, we present Multi-GiLA, the first multilevel force-directed graph visualization algorithm based on a vertex-centric computation paradigm. We implemented Multi-GiLA using the Apache Giraph platform. Experiments show that it can be successfully applied to compute high quality layouts of very large graphs on inexpensive cloud computing platforms.

Abstract 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.

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 drawings of planar graphs and series-parallel graphs. Motivated by this result, we study straight-line drawings of planar graphs where the number of crossings per edge is not bounded by a constant. In this case, we prove that every planar graph admits a straight-line drawing with sub-quadratic area and sub-linear thickness (the thickness of a drawing is the minimum number of colors that can be assigned to the edges so that each color class induces a planar drawing). We also prove that every partial 2-tree (and hence every series-parallel graph) admits a linear-area straight-line drawing with thickness at most 10. It is worth remarking that a drawing with thickness h−1 is h-quasi planar, i.e. it does not contain h-mutually crossing edges. The main ingredient to prove our results is (c, t)-track layouts, a combinatorial tool that can be represented as a drawing where: (i) each vertex is assigned to one of t horizontal layers (tracks), (ii) no two adjacent vertices are on the same track, (iii) each edge receives one of c colors, so that no two edges of the same color (u, v) and (w, z) cross if u, w are on the same track, and v, z are on the same track.

A mixed graph has both directed and undirected edges. We study how to compute a crossing-free drawing of an embedded planar mixed graph, such that it is upward ‘as much as possible’. Roughly speaking, in an upward drawing of a mixed graph all (undirected) edges are monotone in the vertical direction and directed edges flow monotonically from bottom to top according to their orientation. We study quasi-upward drawings of mixed graphs, that is, upward drawings where edges can break the vertical monotonicity in a finite number of edge points, called bends. We describe both efficient heuristic techniques and exact approaches for computing quasi-upward planar drawings of embedded mixed graphs with few bends, and we extensively compare them experimentally: the results suggest that our algorithms are effective in many cases.

A 2-layer drawing of a bipartite graph G is a drawing such that the vertices of each partition set are drawn as points of a distinct horizontal line (called a layer) and the edges are drawn as straight-line segments. We study 2-layer drawings where edges can cross only at right angles; these drawings are called 2-layer right angle crossing drawings (2-layer RAC drawings for short). We focus on the following problem, which we call the maximum 2-layer RAC subgraph (M2LRacS) problem. Given a bipartite graph G, compute a subgraph H of G such that: (i) H admits a 2-layer RAC drawing and (ii) H has the maximum number of edges among the subgraphs of G that satisfy (i). We study this problem both in the no-fixed-layer setting, where no restriction is given on the vertex ordering on each layer, and in the 1-fixed-layer setting, where the ordering of the vertices of one of the two layers is given as part of the input and cannot be changed. The M2LRacS problem is known to be 𝒩𝒫-hard in the no-fixed-layer setting (Di Giacomo, E., Didimo, W., Eades, P. and Liotta, G. (2011) 2-Layer Right Angle Crossing Drawings. Proc. IWOCA 2011, Lecturer Notes in Computer Science 7056, pp. 156–169; Di Giacomo, E., Didimo, W., Eades, P. and Liotta, G. (2014) 2-layer right angle crossing drawings. Algorithmica, 68, 954–997), but no algorithm has been proposed so far to solve it. We prove that the M2LRacS problem remains 𝒩𝒫-hard even in the 1-fixed-layer setting, and provide different heuristics to solve it in the two settings; one of these heuristics is a 3-approximation algorithm for the no-fixed-layer setting. Also, we present the results of an experimental study that compares our heuristics and shows the effectiveness of the 3-approximation algorithm in practice.

A simultaneous geometric embedding (SGE) of two planar graphs |$G_1$| and |$G_2$| with the same vertex set is a pair of straight-line planar drawings |$\Gamma _1$| of |$G_1$| and |$\Gamma _2$| of |$G_2$| such that each vertex is drawn at the same point in |$\Gamma _1$| and |$\Gamma _2$|⁠. Many papers have been devoted to the study of which pairs of graphs admit a SGE, and both positive and negative results have been proved. We extend the study of SGE, by introducing and characterizing a new class of planar graphs that makes it possible to immediately extend several positive results that rely on the property of strictly monotone paths. Moreover, we introduce a relaxation of the SGE setting where |$\Gamma _1$| and |$\Gamma _2$| are required to be quasi-planar (i.e. they can have crossings provided that there are no three mutually crossing edges). This relaxation allows for the simultaneous embedding of pairs of planar graphs that are not simultaneously embeddable in the classical SGE setting and opens up several new interesting research questions.

Fast computation of two-dimensional layouts of hierarchically clustered networks is a well-studied problem in graph visualization. We present algorithmic and experimental advances on the subject: (i) We propose a new drawing algorithm that combines space-filling and fast force-directed methods; it runs in O(nlogn+m) time, where n and m are the number of vertices and edges of the network, respectively. This running time does not depend on the number of clusters, thus the algorithm guarantees good time performances independently of the structure of the cluster hierarchy. As a further advantage, the algorithm can be easily parallelized. (ii) We present an experimental analysis aimed at understanding which clustering algorithms can be used, in combination with our visualization technique, to generate better quality drawings for medium and large networks with small-world and scale-free structure. As far as we know, no previous similar experiments have been done in this respect.

We present a new system, VISFAN, for the visual analysis of financial activity networks. It supports the analyst with effective tools to discover financial crimes, like money laundering and frauds. If compared with other existing systems and methodologies for the analysis of criminal networks, VISFAN presents the following main novelties: (i) It combines bottom-up and top-down interaction paradigms for the visual exploration of complex networks; (ii) It makes it possible to mix automatic and manual clustering; (iii) It allows the analyst to interactively customize the dimensions of each cluster region and to apply different geometric constraints on the layout. VISFAN also implements several tools for social network analysis other than clustering. For example, it computes several indices to measure the centrality of each actor in the network.

Many different approaches have been proposed for the challenging problem of visually analyzing large networks. Clustering is one of the most promising. In this paper, we propose a new clustering technique whose goal is that of producing both intracluster graphs and intercluster graph with desired topological properties. We formalize this concept in the (X,Y) -clustering framework, where Y is the class that defines the desired topological properties of intracluster graphs and X is the class that defines the desired topological properties of the intercluster graph. By exploiting this approach, hybrid visualization tools can effectively combine different node-link and matrix-based representations, allowing users to interactively explore the graph by expansion/contraction of clusters without loosing their mental map. As a proof of concept, we describe the system Visual Hybrid (X,Y)-clustering (VHYXY) that implements our approach and we present the results of case studies to the visual analysis of social networks.

Many different approaches have been proposed for the challenging problem of visually analyzing large networks. Clustering is one of the most promising. In this paper we propose a new goal for clustering that is especially tailored to hybrid-visualization tools. Namely, that of producing both intra-cluster graphs and inter-cluster graph that are suitable for highly-readable visualizations within different representation conventions. We formalize this concept in the (X,Y)-clustering framework, where Y is the class that defines the desired topological properties of intra-cluster graphs and X is the class that defines the desired topological properties of the inter-cluster graph. By exploiting this approach hybrid-visualization tools can effectively combine different node-link and matrix-based representations, allowing the users to interactively explore the graph by expansion/contraction of clusters without loosing their mental map. As a proof of concept, we describe the system VHYXY (Visual Hybrid (X,Y)-clustering) that integrates our techniques and we present the results of case studies to the visual analysis of co-authorship networks.

Systems that support Web site traffic analysis are core business intelligence applications for many companies. Recent papers remark that these systems are especially useful if they measure the users' interest into the relevant concepts described in a Web site rather than counting users' accesses to the distinct pages forming theWeb site. This paper extends existing measures of conceptual Web site traffic analysis and describes a system, called COWA, that supports this analysis by means of network models and graph visualization technologies. The graph drawing algorithmic core of the user interface of COWA is a force directed heuristic that computes a simultaneous embedding of two non-planar graphs. This heuristic optimizes the visualizations in terms of crossing resolution and user's geodesic tendency. Experimental results and case studies show the effectiveness of the proposed approach in practice.

This paper shortly describes a system, called VisForFraud, that uses Information Visualization techniques for the discovery of financial crimes.

A digraph is upward planar if it admits a planar drawing where all edges are monotone in the upward direction. It is known that the problem of testing a digraph for upward planarity is NP-complete in general. This paper describes an $O(n^4)$-time upward planarity testing algorithm for all digraphs that have a series-parallel structure, where n is the number of vertices of the input. This significantly enlarges the family of digraphs for which a polynomial-time testing algorithm is known. Furthermore, the study is extended to general digraphs, and a fixed parameter tractable algorithm for upward planarity testing is described, whose time complexity is $O(d^t \cdot t \cdot n^3 + d \cdot t^2 \cdot n + d^2 \cdot n^2)$ where t is the number of triconnected components of the digraph and d is an upper bound on the diameter of any split component of the digraph. Our results use the new notion of upward spirality that, informally speaking, is a measure of the “level of winding” that a triconnected component of a digraph G can have in an upward planar drawing of G.

The paper describes WhatsOnWeb+, a search clustering engine that allows users to browse and analyze the results of a query by means of enhanced graph visualization techniques. WhatsOnWeb+ integrates a wide array of visual interfaces, animation and interaction functionalities, and clustering technologies. The effectiveness of the different visual interfaces and of the different clustering algorithms implemented in the system has been measured by means of an extensive experimental analysis. The described system represents a significant evolution of a previous clustering engine for the Web.

One of the most challenging issues in mining information from the World Wide Web is the design of systems that present the data to the end user by clustering them into meaningful semantic categories. We show that the analysis of the results of a clustering engine can significantly take advantage of enhanced graph drawing and visualization techniques. We propose a graph-based user interface for Web clustering engines that makes it possible for the user to explore and visualize the different semantic categories and their relationships at the desired level of detail.

Abstract A wide number of practical applications would benefit from automatically generated graphical representations of database schemas, in which tables are represented by boxes, and table attributes correspond to distinct stripes inside each table. Links, connecting attributes of two different tables, represent referential constraints or join relationships, and may attach arbitrarily to the left‐ or to the right‐hand side of the stripes representing the attributes. To our knowledge no drawing technique is available to automatically produce diagrams in such a strongly constrained drawing convention. In this paper we provide a polynomial time algorithm for solving this problem, and test its efficiency and effectiveness against a large test suite. Also, we describe an implementation of a system that uses such an algorithm and we study the main methodological problems we faced in developing such a technology. Copyright © 2002 John Wiley & Sons, Ltd.

We describe a branch-and-bound algorithm for computing an orthogonal grid drawing with the minimum number of bends of a biconnected planar graph. Such algorithm is based on an efficient enumeration schema of the embeddings of a planar graph and on several new methods for computing lower bounds of the number of bends. We experiment such algorithm on a large test suite and compare the results with the state-of-the-art. The experiments show how minimizing the number of bends strongly improves several quality measures of the effectiveness of the drawing. We also present a graphic tool with animation that embodies the algorithm and allows interacting with all the phases of the computation.