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Partial edge drawings (PED) of graphs avoid edge crossings by subdividing each edge into three parts and representing only its stubs, i.e., the parts incident to the end-nodes. The morphing edge drawing model (MED) extends the PED drawing style by animations that smoothly morph each edge between its representation as stubs and the one as a fully drawn segment while avoiding new crossings. Participants of a previous study on MED (Misue and Akasaka, GD19) reported eye straining caused by the animation. We conducted a user study to evaluate how this effect is influenced by varying animation speed and animation dynamic by considering an easing technique that is commonly used in web design. Our results provide indications that the easing technique may help users in executing topology-based tasks accurately. The participants also expressed appreciation for the easing and a preference for a slow animation speed., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

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

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

In a graph story the vertices enter a graph one at a time and each vertex persists in the graph for a fixed amount of time $\omega$, called viewing window. At any time, the user can see only the drawing of the graph induced by the vertices in the viewing window and this determines a sequence of drawings. For readability, we require that all the drawings of the sequence are planar. For preserving the user's mental map we require that when a vertex or an edge is drawn, it has the same drawing for its entire life. We study the problem of drawing the entire sequence by mapping the vertices only to $\omega+k$ given points, where $k$ is as small as possible. We show that: $(i)$ The problem does not depend on the specific set of points but only on its size; $(ii)$ the problem is NP-hard and is FPT when parameterized by $\omega+k$; $(iii)$ there are families of graph stories that can be drawn with $k=0$ for any $\omega$, while for $k=0$ and small values of $\omega$ there are families of graph stories that can be drawn and others that cannot; $(iv)$ there are families of graph stories that cannot be drawn for any fixed $k$ and families of graph stories that require at least a certain $k$., Comment: Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022)

A map graph is a graph admitting a representation in which vertices are nations on a spherical map and edges are shared curve segments or points between nations. We present an explicit fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. The algorithm has time complexity that is linear in the size of the graph and, if the input is a yes-instance, it reports a certificate in the form of a so-called witness. Furthermore, this result is developed within a more general algorithmic framework that allows to test, for any $k$, if the input graph admits a $k$-map (where at most $k$ nations meet at a common point) or a hole-free~$k$-map (where each point of the sphere is covered by at least one nation). We point out that, although bounding the treewidth of the input graph also bounds the size of its largest clique, the latter alone does not seem to be a strong enough structural limitation to obtain an efficient time complexity. In fact, while the largest clique in a $k$-map graph is $\lfloor 3k/2 \rfloor$, the recognition of $k$-map graphs is still open for any fixed $k \ge 5$.

We study the parameterized complexity of the $s$-Club Cluster Edge Deletion problem: Given a graph $G$ and two integers $s \ge 2$ and $k \ge 1$, is it possible to remove at most $k$ edges from $G$ such that each connected component of the resulting graph has diameter at most $s$? This problem is known to be NP-hard already when $s = 2$. We prove that it admits a fixed-parameter tractable algorithm when parameterized by $s$ and the treewidth of the input graph.

This paper studies the problem of computing quasi-upward planar drawings of bimodal plane digraphs with minimum curve complexity, i.e., drawings such that the maximum number of bends per edge is minimized. We prove that every bimodal plane digraph admits a quasi-upward planar drawing with curve complexity two, which is worst-case optimal. We also show that the problem of minimizing the curve complexity in a quasi-upward planar drawing can be modeled as a min-cost flow problem on a unit-capacity planar flow network. This gives rise to an $\tilde{O}(m^\frac{4}{3})$-time algorithm that computes a quasi-upward planar drawing with minimum curve complexity; in addition, the drawing has the minimum number of bends when no edge can be bent more than twice. For a contrast, we show bimodal planar digraphs whose bend-minimum quasi-upward planar drawings require linear curve complexity even in the variable embedding setting., Comment: Appears in the Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021)

Hybrid visualizations mix different metaphors in a single layout of a network. In particular, the popular NodeTrix model, introduced by Henry, Fekete, and McGuffin in 2007, combines node-link diagrams and matrix-based representations to support the analysis of real-world networks that are globally sparse but locally dense. That idea inspired a series of works, proposing variants or alternatives to NodeTrix. We present a user study that compares the classical node-link model and three hybrid visualization models designed to work on the same types of networks. The results of our study provide interesting indications about advantages/drawbacks of the considered models on performing classical tasks of analysis. At the same time, our experiment has some limitations and opens up to further research on the subject., Comment: Appears in the Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021)

Storyline visualizations depict the temporal dynamics of social interactions, as they describe how groups of actors (individuals or organizations) change over time. A common constraint in storyline visualizations is that an actor cannot belong to two different groups at the same time instant. However, this constraint may be too severe in some application scenarios, thus we generalize the model by allowing an actor to simultaneously belong to distinct groups at any point in time. We call this model Storyline with Ubiquitous Actors (SUA). Essential to our model is that an actor is represented as a tree rather than a single line. We describe an algorithmic pipeline to compute storyline visualizations in the SUA model and discuss case studies on publication data., Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020)

In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings with two and three available colors and present improved bounds on the size of the monochromatic connected components for two meaningful subclasses of planar graphs, namely maximal outerplanar graphs and complete planar 3-trees.

We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, \dots, G_k$, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each $G_i$ is a tree and $k=3$. We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with $n \geq 12$ vertices admits a 1-planar packing, while such a packing does not exist if $n \leq 10$.

We study the interplay between embedding constrained planarity and hybrid planarity testing. We consider a constrained planarity testing problem, called 1-Fixed Constrained Planarity, and prove that this problem can be solved in quadratic time for biconnected graphs. Our solution is based on a new definition of fixedness that makes it possible to simplify and extend known techniques about Simultaneous PQ-Ordering. We apply these results to different variants of hybrid planarity testing, including a relaxation of NodeTrix Planarity with fixed sides, that allows rows and columns to be independently permuted.

Many real-world networks are globally sparse but locally dense. Typical examples are social networks, biological networks, and information networks. This double structural nature makes it difficult to adopt a homogeneous visualization model that clearly conveys an overview of the network and the internal structure of its communities at the same 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 paper 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 is intuitive and 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, and illustrate case studies on real-world networks., Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019)

Hierarchical embedding constraints define a set of allowed cyclic orders for the edges incident to the vertices of a graph. These constraints are expressed in terms of FPQ-trees. FPQ-trees are a variant of PQ-trees that includes F-nodes in addition to P- and to Q-nodes. An F-node represents a permutation that is fixed, i.e., it cannot be reversed. Let $G$ be a graph such that every vertex of $G$ is equipped with a set of FPQ-trees encoding hierarchical embedding constraints for its incident edges. We study the problem of testing whether $G$ admits a planar embedding such that, for each vertex $v$ of $G$, the cyclic order of the edges incident to $v$ is described by at least one of the FPQ-trees associated with~$v$. We prove that the problem is fixed-parameter tractable for biconnected graphs, where the parameters are the treewidth of $G$ and the number of FPQ-trees associated with every vertex of $G$. We also show that the problem is NP-complete if parameterized by the number of FPQ-trees only, and W[1]-hard if parameterized by the treewidth only. Besides being interesting on its own right, the study of planarity testing with hierarchical embedding constraints can be used to address other planarity testing problems. In particular, we apply our techniques to the study of NodeTrix planarity testing of clustered graphs. We show that NodeTrix planarity testing with fixed sides is fixed-parameter tractable when parameterized by the size of the clusters and by the treewidth of the multi-graph obtained by collapsing the clusters to single vertices, provided that this graph is biconnected.

We study the question whether a crossing-free 3D morph between two straight-line drawings of an $n$-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with $O(\log n)$ steps, while for the latter $\Theta(n)$ steps are always sufficient and sometimes necessary., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $h$-Clique2Path Planarity: Given a graph $G$, whose vertices are partitioned into subsets of size at most $h$, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of $G$ is planar. We study this problem when $G$ is a simple topological graph, and establish its complexity in relation to $k$-planarity. We prove that $h$-Clique2Path Planarity is NP-complete even when $h=4$ and $G$ is a simple $3$-plane graph, while it can be solved in linear time, for any $h$, when $G$ is $1$-plane., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. The existence of greedy drawings has been widely studied under different topological and geometric constraints, such as planarity, face convexity, and drawing succinctness. We introduce greedy rectilinear drawings, in which each edge is either a horizontal or a vertical segment. These drawings have several properties that improve human readability and support network routing. We address the problem of testing whether a planar rectilinear representation, i.e., a plane graph with specified vertex angles, admits vertex coordinates that define a greedy drawing. We provide a characterization, a linear-time testing algorithm, and a full generative scheme for universal greedy rectilinear representations, i.e., those for which every drawing is greedy. For general greedy rectilinear representations, we give a combinatorial characterization and, based on it, a polynomial-time testing and drawing algorithm for a meaningful subset of instances., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

An ortho-polygon visibility representation $\Gamma$ of a $1$-plane graph $G$ (OPVR of $G$) is an embedding preserving drawing that maps each vertex of $G$ to a distinct orthogonal polygon and each edge of $G$ to a vertical or horizontal visibility between its end-vertices. The representation $\Gamma$ has vertex complexity $k$ if every polygon of $\Gamma$ has at most $k$ reflex corners. It is known that $3$-connected $1$-plane graphs admit an OPVR with vertex complexity at most twelve, while vertex complexity at least two may be required in some cases. In this paper, we reduce this gap by showing that vertex complexity five is always sufficient, while vertex complexity four may be required in some cases. These results are based on the study of the combinatorial properties of the B-, T-, and W-configurations in $3$-connected $1$-plane graphs. An implication of the upper bound is the existence of a $\tilde{O}(n^\frac{10}{7})$-time drawing algorithm that computes an OPVR of an $n$-vertex $3$-connected $1$-plane graph on an integer grid of size $O(n) \times O(n)$ and with vertex complexity at most five., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph $G = (V,E)$ is $(k,p)$-planar if $V$ can be partitioned into clusters of size at most $k$ such that $G$ admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region; (ii) cluster regions are pairwise disjoint, (iii) each vertex $v \in V$ is identified with at most $p$ distinct points, called \emph{ports}, on the boundary of its cluster region; (iv) each inter-cluster edge $(u,v) \in E$ is identified with a Jordan arc connecting a port of $u$ to a port of $v$; (v) inter-cluster edges do not cross or intersect cluster regions except at their endpoints. We first tightly bound the number of edges in a $(k,p)$-planar graph with $p

The assignment of weights to attacks in a classical Argumentation Framework allows to compute semantics by taking into account the different importance of each argument. We represent a Weighted Argumentation Framework by a non-binary matrix, and we characterize the basic extensions (such as w-admissible, w- stable, w-complete) by analysing sub-blocks of this matrix. Also, we show how to reduce the matrix into another one of smaller size, that is equivalent to the original one for the determination of extensions. Furthermore, we provide two algorithms that allow to build incrementally w-grounded and w-preferred extensions starting from a w-admissible extension.

We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant $k$. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be chosen arbitrarily. We show that NodeTrix planarity testing with fixed sides can be solved in $O(k^{3k+\frac{3}{2}} \cdot n)$ time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in $O(n)$ time for $k = 2$, but it is NP-complete for any $k > 2$. NodeTrix planarity testing remains NP-complete also in the free sides model when $k > 4$., Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017)