• In this paper we study a variant of Vertex Cover where the activities of vertices are characterized by time intervals. We explore a scenario where the temporal span of each vertex’s activity interval is bounded by an integer, and the objective is to maximize the number of (temporal) edges that are covered. • We establish the APX-hardness of this problem and the NP-hardness of the corresponding decision problem, even under the restricted conditions where: the temporal domain comprises only two timestamps and each edge appears at most once and; no two edges are associated to a same label. • We delve into the parameterized complexity of the problem, offering two fixed-parameter algorithms parameterized by: the number k of temporal edges covered by the solution, and the number h of temporal edges left uncovered by the solution. • We focus again on the approximability of the problem and present a polynomial-time approximation algorithm achieving a factor of 3 4 . Different variants of Vertex Cover have recently garnered attention in the context of temporal graphs. One of these variants is motivated by the need to summarize timeline activities in social networks. Here, the activities of individual vertices, representing users, are characterized by time intervals. In this paper, we explore a scenario where the temporal span of each vertex’s activity interval is bounded by an integer ℓ, and the objective is to maximize the number of (temporal) edges that are covered. We establish the APX-hardness of this problem and the NP-hardness of the corresponding decision problem, even under the restricted conditions where: the temporal domain comprises only two timestamps and each edge appears at most once and; no two edges are associated to a same label. Subsequently, we delve into the parameterized complexity of the problem, offering two fixed-parameter algorithms parameterized by: (i) the number k of temporal edges covered by the solution, and (ii) the number h of temporal edges not covered by the solution. Finally, we present a polynomial-time approximation algorithm achieving a factor of 3 4 .

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.

Problem solving is a composite cognitive process, invoking a number of cognitive mechanisms, such as perception and memory. Individuals may form collectives to solve a given problem together in collaboration, especially when complexity is perceived to be high. To determine if and when collaborative problem solving is desired in the context of visual graph analysis, we compare ad hoc pairs to individuals and nominal pairs, when solving different tasks in mixed reality. We discuss the results of an experiment with 72 participants performed in two countries and three languages. We apply the concept of task instance complexity to quantify the visual demand of tasks used in the experiment. Our results show the importance of using nominal groups as a benchmark for evaluating collaborative virtual environments. We conclude that 3D graph representation is not sufficient to induce better collaborative results compared to the benchmark.

Storyline visualizations represent character interactions over time. When these characters belong to different groups, a new research question emerges: how can we balance optimization of readability across the groups while preserving the overall narrative structure of the story? Traditional algorithms that optimize global readability metrics (like minimizing crossings) can introduce quality biases between the different groups based on their cardinality and other aspects of the data. Visual consequences of these biases are: making characters of minority groups disproportionately harder to follow, and visually deprioritizing important characters when their curves become entangled with numerous secondary characters. We present F2Stories, a modular framework that addresses these challenges in storylines by offering three complementary optimization modes: (1) fairnessMode ensures that no group bears a disproportionate burden of visualization complexity regardless of their representation in the story; (2) focusMode allows prioritizing a group of characters while maintaining good readability for secondary characters; and (3) standardMode globally optimizes classical aesthetic metrics. Our approach is based on Mixed Integer Linear Programming (MILP), offering optimality guarantees, precise balancing of competing metrics through weighted objectives, and the flexibility to incorporate complex fairness concepts as additional constraints without the need to redesign the entire algorithm. We conducted an extensive experimental analysis to demonstrate how F2Stories enables more fair or focus group-prioritized storyline visualizations while maintaining adherence to established layout constraints. Our evaluation includes comprehensive results from a detailed case study that shows the effectiveness of our approach in real-world narrative contexts. An open access copy of this paper and all supplemental materials are available at osf.io/e2qvy.

A decision tree recursively splits a feature space R d and then assigns class labels based on the resulting partition. Decision trees have been part of the basic machine-learning toolkit for decades. A large body of work considers heuristic algorithms that compute a decision tree from training data, usually aiming to minimize in particular the size of the resulting tree. In contrast, little is known about the complexity of the underlying computational problem of computing a minimum-size tree for the given training data. We study this problem with respect to the number d of dimensions of the feature space R d , which contains n training examples. We show that it can be solved in O ( n 2 d + 1 ) time, but under reasonable complexity-theoretic assumptions it is not possible to achieve f ( d ) ⋅ n o ( d / log ⁡ d ) running time. The problem is solvable in ( d R ) O ( d R ) ⋅ n 1 + o ( 1 ) time if there are exactly two classes and R is an upper bound on the number of tree leaves labeled with the first class.

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.

Motivated by the need for decision-making systems that avoid bias and discrimination, the concept of fairness recently gained traction in the broad field of artificial intelligence , stimulating new research also within the information visualization community. In this paper, we introduce a notion of fairness in network visualization, specifically for orthogonal and for straight-line drawings of graphs, two foundational paradigms in the field. We investigate the following research questions: (i) What is the price, in terms of global readability , of incorporating fairness constraints in graph drawings? (ii) How unfair is a graph drawing that does not optimize fairness as a primary objective ? We present both theoretical and empirical results. In particular, we design and implement two optimization algorithms for multi-objective functions, one based on an ILP model for orthogonal drawings, and one based on gradient descent for straight-line drawings. In a nutshell, we experimentally show that it is possible to significantly increase the fairness of a drawing by paying a relatively small amount in terms of reduced global readability. Also, we present a use case in which we qualitatively evaluate our approach on a practical scenario.

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.

Graph and network visualization supports exploration, analysis and communication of relational data arising in many domains: from biological and social networks, to transportation and powergrid systems. With the arrival of AI-based question-answering tools, issues of trustworthiness and explainability of generated answers motivate a significant new role for visualization. In the context of graphs, we see the need for visualizations that can convince a critical audience that an assertion (e. g., from an AI) about the graph under analysis is valid. The requirements for such representations that convey precisely one specific graph property are quite different from standard network visualization criteria which optimize general aesthetics and readability. In this paper, we aim to provide a comprehensive introduction to visual proofs of graph properties and a foundation for further research in the area. We present a framework that defines what it means to visually prove a graph property. In the process, we introduce the notion of a visual certificate, that is, a specialized faithful graph visualization that leverages the viewer's perception, in particular, pre-attentive processing (e. g., via pop-out effects), to verify a given assertion about the represented graph. We also discuss the relationships between visual complexity, cognitive load and complexity theory, and propose a classification based on visual proof complexity. Then, we provide further examples of visual certificates for problems in different visual proof complexity classes. Finally, we conclude the paper with a discussion of the limitations of our model and some open problems.

Graph and network visualization supports exploration, analysis and communication of relational data arising in many domains: from biological and social networks, to transportation and powergrid systems. With the arrival of AI-based question-answering tools, issues of trustworthiness and explainability of generated answers motivate a greater role for visualization. In the context of graphs, we see the need for visualizations that can convince a critical audience that an assertion about the graph under analysis is valid. The requirements for such representations that convey precisely one specific graph property are quite different from standard network visualization criteria which optimize general aesthetics and readability. In this paper, we aim to provide a comprehensive introduction to visual proofs of graph properties and a foundation for further research in the area. We present a framework that defines what it means to visually prove a graph property. In the process, we introduce the notion of a visual certificate, that is, a specialized faithful graph visualization that leverages the viewer's perception, in particular, pre-attentive processing (e. g. via pop-out effects), to verify a given assertion about the represented graph. We also discuss the relationships between visual complexity, cognitive load and complexity theory, and propose a classification based on visual proof complexity. Finally, we provide examples of visual certificates for problems in different visual proof complexity classes.

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 A map is a partition of the sphere into interior-disjoint regions homeomorphic to closed disks. Some regions are labeled as nations, while the remaining ones are labeled as holes. A map in which at mostknations touch at the same point is ak-map, while it is hole-free if it contains no holes. A graph is a map graph if there is a bijection between its vertices and the nations of a map, such that two nations touch if and only the corresponding vertices are connected by an edge. We present a fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. Its time complexity is linear in the size of the graph. It reports a certificate in the form of a so-called witness, if the input is a yes-instance. Our algorithmic framework is general enough to test, for anyk, if the input graph admits ak-map or a hole-free k-map.

Abstract The planar slope number $${{\,\textrm{psn}\,}}(G)$$ psn ( G ) of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that $${{\,\textrm{psn}\,}}(G) \in O(c^{\Delta })$$ psn ( G ) ∈ O ( c Δ ) for every planar graph G of maximum degree $$\Delta $$ Δ . This upper bound has been improved to $$O(\Delta ^5)$$ O ( Δ 5 ) if G has treewidth three, and to $$O(\Delta )$$ O ( Δ ) if G has treewidth two. In this paper we prove $${{\,\textrm{psn}\,}}(G) \le \max \{4,\Delta \}$$ psn ( G ) ≤ max { 4 , Δ } when G is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that $$O(\Delta ^2)$$ O ( Δ 2 ) slopes suffice for nested pseudotrees.

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 .

We study the problem of representing a graph as a storyplan, a recently introduced model for dynamic graph visualization. It is based on a sequence of frames, each showing a subset of vertices and a planar drawing of their induced subgraphs, where vertices appear and disappear over time. Namely, in the StoryPlan problem, we are given a graph and we want to decide whether there exists a total vertex appearance order for which a storyplan exists. We prove that the problem is NP-complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of the input graph. We prove that partial 3-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP-complete if the vertex appearance order is given and we have to choose how to draw the frames.

In this work, we continue the study of the area required for convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally require the obtained drawings to have bounded edge-vertex resolution, that is, the closest distance between a vertex and any non-incident edge in the drawing is lower bounded by a constant that does not depend on the size of the graph. We present a drawing algorithm that takes as input a 3-connected plane graph with n vertices and f internal faces, and computes a convex straight-line drawing with edge-vertex resolution at least 12 on an integer grid of size (n−2+a)×(n−2+a), where a=min⁡{n−3,f}. Our result improves the previously best-known area bound of (3n−7)×(3n−7)/2 by Chrobak, Goodrich and Tamassia.

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.

A decision tree recursively splits a feature space \mathbb{R}^d and then assigns class labels based on the resulting partition. Decision trees have been part of the basic machine-learning toolkit for decades. A large body of work considers heuristic algorithms that compute a decision tree from training data, usually aiming to minimize in particular the size of the resulting tree. In contrast, little is known about the complexity of the underlying computational problem of computing a minimum-size tree for the given training data. We study this problem with respect to the number d of dimensions of the feature space \mathbb{R}^d, which contains n training examples. We show that it can be solved in O(n^(2d + 1)) time, but under reasonable complexity-theoretic assumptions it is not possible to achieve f(d) * n^o(d / log d) running time. The problem is solvable in (dR)^O(dR) * n^(1+o(1)) time, if there are exactly two classes and R is an upper bound on the number of tree leaves labeled with the first class.

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions.

The problem of deciding whether a biconnected planar digraph $G=(V,E)$ can be augmented to become an $st$-planar graph by adding a set of oriented edges $E' \subseteq V \times V$ is known to be NP-complete. We show that the problem is fixed-parameter tractable when parameterized by the size of the set $E'$.

Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an $st$-orientation orients each edge of the input graph such that the resulting digraph is acyclic, and it contains a single source $s$ and a single sink $t$. Computing an $st$-orientation of a graph can be done efficiently, and it finds notable applications in graph algorithms and in particular in graph drawing. On the other hand, finding an $st$-orientation with at most $k$ transitive edges is more challenging and it was recently proven to be NP-hard already when $k=0$. We strengthen this result by showing that the problem remains NP-hard even for graphs of bounded diameter, and for graphs of bounded vertex degree. These computational lower bounds naturally raise the question about which structural parameters can lead to tractable parameterizations of the problem. Our main result is a fixed-parameter tractable algorithm parameterized by treewidth.

We study the parameterized complexity of the s-Club Cluster Edge Deletion (s-Club Cluster Vertex Deletion) problem: Given a graph G and two integers s≥2 and k≥1, is it possible to remove at most k edges (vertices) from G such that each connected component of the resulting graph has diameter at most s? Both s-Club Cluster Edge Deletion and s-Club Cluster Vertex Deletion problems are known to be NP-hard already when s=2. We prove that they admit a fixed-parameter tractable algorithm when parameterized by s and the treewidth of the input graph. The proof is based on a unified algorithm that solves the more general problem in which both edges and vertices can be removed from the input graph to obtain a set of disjoint components with bounded diameter. Our approach can also be exploited to solve a related problem, namely s-Club Cover, which asks whether it is possible to cover the vertices of a graph with at most d different s-clubs, for some fixed d≥1 and s≥2.

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.

An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs.

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

A $k$-page book embedding of a graph $G$ draws the vertices of $G$ on a line and the edges on $k$ half-planes (called pages) bounded by this line, such that no two edges on the same page cross. We study the problem of determining whether $G$ admits a $k$-page book embedding both when the linear order of the vertices is fixed, called ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$, or not fixed, called ${\rm B{\small OOK}~T{\small HICKNESS}}$. Both problems are known to be ${\sf NP}$-complete in general. We show that ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$ and ${\rm B{\small OOK}~T{\small HICKNESS}}$ are fixed-parameter tractable parameterized by the vertex cover number of the graph and that ${\rm F{\small IXED}-O{\small RDER}~B{\small OOK}~T{\small HICKNESS}}$ is fixed-parameter tractable parameterized by the pathwidth of the vertex order.

A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg states that the queue number of planar graphs is bounded.This conjecture has been partially settled in the positive for several sub- families of planar graphs (most of which have bounded treewidth).

A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg [ SIAM J. Discrete Math., 5 (1992), pp. 398--412] states that the queue number of planar graphs is bounded. This conjecture has been partially settled in the positive for several subfamilies of planar graphs (most of which have bounded treewidth). In this paper, we make a further important step towards settling this conjecture. We prove that planar graphs of bounded degree (which may have unbounded treewidth) have bounded queue number. A notable implication of this result is that every planar graph of bounded degree admits a three-dimensional straight-line grid drawing in linear volume. Further implications are that every planar graph of bounded degree has bounded track number, and that every $k$-planar graph (i.e., every graph that can be drawn in the plane with at most $k$ crossings per edge) of bounded degree has bounded queue number.

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.

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi [Therese C. Biedl and Saeed Mehrabi, 2017] studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the L_1 (Manhattan) shortest paths in the plane that avoid obstacles. In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in R^d as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.

Let G be a simple topological graph and let Gamma be a polyline drawing of G. We say that Gamma partially preserves the topology of G if it has the same external boundary, the same rotation system, and the same set of crossings as G. Drawing Gamma fully preserves the topology of G if the planarization of G and the planarization of Gamma have the same planar embedding. We show that if the set of crossing-free edges of G forms a connected spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Omega(sqrt{n}). Concerning drawings that fully preserve the topology, we show that if G has skewness k, it admits one such drawing with curve complexity at most 2k; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

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.

A (weak) rectangle visibility representation, or simply an RVR, of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Given a graph with a fixed embedding in the plane, we show that the problem of testing whether this graph has an embedding-preserving RVR can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs, i.e., for embedded graphs having at most one crossing per edge. The linear time algorithm uses three forbidden configurations, which extend the set known for straight-line drawings of 1-plane graphs. The algorithm first checks for the presence of these forbidden configurations in the input graph, and then either an embedding-preserving RVR is computed (also in linear time) or a forbidden configuration is reported as a negative witness. Finally, we discuss extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1).

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 rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

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.