We introduce Clusterix, a novel hybrid visualization model for representing hierarchically clustered networks, which also supports directed and weighted edges. Clusterix 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 Clusterix, 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 Clusterix and evaluated its performance through case studies and qualitative feedback from experts in different application domains.

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.

Graph drawing aesthetics have traditionally been optimized for human readers, leading to well-established principles such as reducing edge crossings, enhancing symmetry, and minimizing bends. These criteria shape layout algorithms and define what "readability" means in network visualization. Today, however, visualizations are increasingly interpreted not only by humans but also by Large Language Models, which now routinely process scientific papers, blog posts, and figures found online and in the wild. This raises a fundamental question: do the same aesthetic criteria that benefit humans also support AI-based visual understanding? Exploring these criteria would inform us, visualization designers and researchers, on how to create network visualizations that are fit for both human and AI readers, and, in turn, enable AI to navigate visualizations in the wild properly. In order to study these criteria, we replicated two foundational studies that established graph drawing aesthetics for humans-only, in our case, the "participant" is AI, and not human. Our findings indicate that AI and human preferences for network visualization are not necessarily the same. We offer a first glance at AI-specific readability criteria, and point toward visualization guidelines that support both human and machine interpretation.

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; (ii) it can be solved in linear time when an upward planar embedding of G is fixed; (iii) 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; (iv) 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. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Linear layouts of graphs – particularly stack and queue layouts – are well-established types of representations in graph drawing, thanks to their connection with numerous theoretical and practical problems. In such layouts, all vertices are linearly ordered and the edges are partitioned into sets that avoid specific forbidden configurations: in a stack layout no two independent edges within the same set cross, whereas in a queue layout no two independent edges within the same set are nested. A central problem in this context is to determine, for a given graph G, its stack number or queue number, that is, the minimum number of sets into which the edges can be partitioned so that a corresponding stack or queue layout of G exists. In this work, we introduce a relaxation of stack and queue layouts, which allows some forbidden patterns for the edges in the same set. Namely, for a given integer k > 0, a k-defective stack layout (resp. a k-defective queue layout) allows an edge to be in a crossing (resp. nesting) relationship with at most k edges within the same set. Our motivation is to extend the classes of graphs that admit linear layouts using a limited number of edge-partition sets, at the cost of allowing some defects. We study defective linear layouts both from a combinatorial and from an algorithmic perspective, providing an array of results across different graph classes and parameters.

A planar orthogonal drawing Γ of a connected planar graph G is a geometric representation of G such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and no two edges intersect except at common end-points. A bend of Γ is a point of an edge where a horizontal and a vertical segment meet. Drawing Γ is bend-minimum if it has the minimum number of bends over all possible planar orthogonal drawings of G. Its curve complexity is the maximum number of bends per edge. In this paper we present a linear-time algorithm for the computation of planar orthogonal drawings of 3-graphs (i.e., graphs with vertex-degree at most three), that minimizes both the total number of bends and the curve complexity. The algorithm works in the so-called variable embedding setting, that is, it can choose among the exponentially many planar embeddings of the input graph. While the time complexity of minimizing the total number of bends of a planar orthogonal drawing of a 3-graph in the variable embedding settings is a long standing, widely studied, open question, the existence of an orthogonal drawing that is optimal both in the total number of bends and in the curve complexity was previously unknown. Our result combines several graph decomposition techniques, novel data-structures, and efficient approaches to re-rooting decomposition trees.

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.

We present an interactive visual analytics system, called VA-CHIP, integrated into a Travel Recommender System (TRS) called CHIP. A TRS suggests cultural itineraries specifically tailored to the travelers’ preferences expressed in terms of a set of Topics of Interest (TOIs). To construct the itineraries to recommend, CHIP describes the cultural attractions (Points of Interest – POIs) by means of a set of relevance scores with respect to the same set of TOIs. These descriptions of the POIs are a valuable source of information for territorial administrators (policymakers), who can carry out a range of analytical tasks based on them. To support this process, VA-CHIP enables multi-perspective reasoning on the POIs and TOIs datasets through a suite of coordinated multiple views, which include a semantic similarity network, a dimensionality-reduction-based visualization, and a geographic view.

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.

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.

Computing a Euclidean minimum spanning tree of a set of points is a seminal problem in computational geometry and geometric graph theory. We combine it with another classical problem in graph drawing, namely computing a monotone geometric representation of a given graph. More formally, given a finite set S of points in the plane and a finite set 𝒟 of directions, a geometric spanning tree T with vertex set S is 𝒟 -monotone if, for every pair {u,v} of vertices of T, there exists a direction d ∈𝒟 for which the unique path from u to v in T is monotone with respect to d. We provide a characterization of 𝒟 -monotone spanning trees. Based on it, we show that a 𝒟 -monotone spanning tree of minimum length can be computed in polynomial time if the number k=|𝒟| of directions is fixed, both when (i) the set 𝒟 of directions is prescribed and when (ii) the objective is to find a minimum-length 𝒟 -monotone spanning tree over all sets 𝒟 of k directions. For k = 2 , we describe algorithms that are much faster than those for the general case. Furthermore, in contrast to the classical Euclidean minimum spanning tree, whose vertex degree is at most six, we show that for every even integer k, there exists a point set S_k and a set 𝒟 of k directions such that any minimum-length 𝒟 -monotone spanning tree of S_k has maximum vertex degree 2k.

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.

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

A drawing F of a graph G 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 k, we say that F is a k+-real face drawing of G if the boundary of each face of F contains at least k vertices of G. Graphs that admit a k+-real face drawing are k+-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 k+-real face graphs, that is, the complexity of testing whether a given graph is k+-real face, for desired values of k. 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.

This paper presents a new decision support system offered for an in-depth analysis of semantic networks, which can provide insights for a better exploration of a brand's image and the improvement of its connectivity. In terms of network analysis, we show that this goal is achieved by solving an extended version of the Maximum Betweenness Improvement problem, which includes the possibility of considering adversarial nodes, constrained budgets, and weighted networks - where connectivity improvement can be obtained by adding links or increasing the weight of existing connections. Our contribution includes a new algorithmic framework and the integration of this framework into a software system called Brand Network Booster (BNB), which supports brand connectivity evaluation and improvement. We present this new system together with three case studies, and we also discuss its performance. Our tool and approach are valuable to both network scholars and in facilitating strategic decision-making processes for marketing and communication managers across various sectors, be it public or private.

Computing planar orthogonal drawings with the minimum number of bends is one of the most relevant topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001). From the parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number of bends, the number of vertices of degree at most two, and the treewidth of the input graph (Di Giacomo et al., 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by the number of vertices of degree at most two plus the number of bends. As a consequence, rectilinear planarity testing lies in FPT parameterized by the number of vertices of degree at most two.

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 point.

Graph drawing beyond planarity is a research area that has received an increasing attention in the last twenty years, driven by the necessity to mitigate the visual complexity inherent in geometric representations of non-planar graphs. This research area stems from the study of graph layouts with forbidden crossing configurations, a well-established subject in geometric and topological graph theory. In this context, the contribution of this paper is as follows: 1) We introduce a new hierarchy of graph families, called k(+) -real face graphs; for any integer k >= 1 , a graph G is a k(+) -real face graph if it admits a drawing Gamma in the plane such that the boundary of each face (formed by vertices, crossings, and edges) contains at least k vertices of G (" k(+) " stands for k or more); 2) We give tight upper bounds on the edge density of k(+) -real face graphs, namely we prove that n-vertex 1+ -real face and 2+ -real face graphs have at most 5n-10 and 4n-8 edges, respectively. Furthermore, in a constrained scenario in which all vertices must lie on the boundary of the external face, 1+ -real face and 2+ -real face graphs have at most 3n-6 and 2.5n-4 edges, respectively; 3) We characterize the complete graphs that admit a k(+) -real face drawing or an outer k(+) -real face drawing for any k >= 1 . We also provide a clear picture for the majority of complete bipartite graphs; and 4) We establish relationships between k(+) -real face graphs and other prominent beyond-planar graph families; notably, we show that for any k >= 1 , the class of k(+) -real face graphs is not included in any family of beyond-planar graphs with hereditary property.