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Problem solving is a composite cognitive process, invoking a number of systems and subsystems, such as perception and memory. Individuals may form collectives to solve a given problem together, in collaboration, especially when complexity is thought to be high. To determine if and when collaborative problem solving is desired, we must quantify collaboration first. For this, we investigate the practical virtue of collaborative problem solving. Using visual graph analysis, we perform a study with 72 participants in two countries and three languages. We compare ad hoc pairs to individuals and nominal pairs, solving two different tasks on graphs in visuospatial mixed reality. The average collaborating pair does not outdo its nominal counterpart, but it does have a significant trade-off against the individual: an ad hoc pair uses 1.46 more time to achieve 4.6 higher accuracy. We also use the concept of task instance complexity to quantify differences in complexity. As task instance complexity increases, these differences largely scale, though with two notable exceptions. With this study we show the importance of using nominal groups as benchmark in collaborative virtual environments research. We conclude that a mixed reality environment does not automatically imply superior collaboration., Comment: 18 pages, 7 figures

In the advancement of industrial informatization, Unsupervised Industrial Anomaly Detection (UIAD) technology effectively overcomes the scarcity of abnormal samples and significantly enhances the automation and reliability of smart manufacturing. While RGB, 3D, and multimodal anomaly detection have demonstrated comprehensive and robust capabilities within the industrial informatization sector, existing reviews on industrial anomaly detection have not sufficiently classified and discussed methods in 3D and multimodal settings. We focus on 3D UIAD and multimodal UIAD, providing a comprehensive summary of unsupervised industrial anomaly detection in three modal settings. Firstly, we compare our surveys with recent works, introducing commonly used datasets, evaluation metrics, and the definitions of anomaly detection problems. Secondly, we summarize five research paradigms in RGB, 3D and multimodal UIAD and three emerging industrial manufacturing optimization directions in RGB UIAD, and review three multimodal feature fusion strategies in multimodal settings. Finally, we outline the primary challenges currently faced by UIAD in three modal settings, and offer insights into future development directions, aiming to provide researchers with a thorough reference and offer new perspectives for the advancement of industrial informatization. Corresponding resources are available at https://github.com/Sunny5250/Awesome-Multi-Setting-UIAD., Comment: 28 pages, 18 figures

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., Comment: Appears in the Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly $y$-monotone curve. A graph is $s$-span weakly leveled planar if it admits such a drawing where the edges have span at most $s$; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing $s$-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter $s$ and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of $2$-outerplanar graphs generalizing Halin graphs, are $\Theta(\log n)$-span weakly leveled planar and $4$-span weakly leveled planar when $3$-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration., Comment: Appears in the Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k \ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that upward-$k$-planarity testing is NP-complete already for $k =1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward $1$-planar drawing in which all vertices are incident to the outer face., Comment: This is the extended version, with full appendix, of a paper to appear in the Proc. 32nd Int. Symp. Graph Drawing & Network Visualization (GD 2024)

A Collaborative Artificial Intelligence System (CAIS) performs actions in collaboration with the human to achieve a common goal. CAISs can use a trained AI model to control human-system interaction, or they can use human interaction to dynamically learn from humans in an online fashion. In online learning with human feedback, the AI model evolves by monitoring human interaction through the system sensors in the learning state, and actuates the autonomous components of the CAIS based on the learning in the operational state. Therefore, any disruptive event affecting these sensors may affect the AI model's ability to make accurate decisions and degrade the CAIS performance. Consequently, it is of paramount importance for CAIS managers to be able to automatically track the system performance to understand the resilience of the CAIS upon such disruptive events. In this paper, we provide a new framework to model CAIS performance when the system experiences a disruptive event. With our framework, we introduce a model of performance evolution of CAIS. The model is equipped with a set of measures that aim to support CAIS managers in the decision process to achieve the required resilience of the system. We tested our framework on a real-world case study of a robot collaborating online with the human, when the system is experiencing a disruptive event. The case study shows that our framework can be adopted in CAIS and integrated into the online execution of the CAIS activities., Comment: This paper is accepted at the 3rd International Conference on AI Engineering - Software Engineering for AI (CAIN 2024), Lisbon, Portugal

The \emph{local edge-length ratio} of a planar straight-line drawing $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$ that share a common vertex. The \emph{global edge-length ratio} of $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$. The local (global) edge-length ratio of a planar graph is the infimum over all local (global) edge-length ratios of its planar straight-line drawings. We show that there exist planar graphs with $n$ vertices whose local edge-length ratio is $\Omega(\sqrt{n})$. We then show a technique to establish upper bounds on the global (and hence local) edge-length ratio of planar graphs and~apply~it to Halin graphs and to other families of graphs having outerplanarity two.

We study the problem of gradually representing a complex graph as a sequence of drawings of small subgraphs whose union is the complex graph. The sequence of drawings is called \emph{storyplan}, and each drawing in the sequence is called a \emph{frame}. In an outerplanar storyplan, every frame is outerplanar; in a forest storyplan, every frame is acyclic. We identify graph families that admit such storyplans and families for which such storyplans do not always exist. In the affirmative case, we present efficient algorithms that produce straight-line storyplans., Comment: Appears in Proc. SOFSEM 2024

A Collaborative Artificial Intelligence System (CAIS) works with humans in a shared environment to achieve a common goal. To recover from a disruptive event that degrades its performance and ensures its resilience, a CAIS may then need to perform a set of actions either by the system, by the humans, or collaboratively together. As for any other system, recovery actions may cause energy adverse effects due to the additional required energy. Therefore, it is of paramount importance to understand which of the above actions can better trade-off between resilience and greenness. In this in-progress work, we propose an approach to automatically evaluate CAIS recovery actions for their ability to trade-off between the resilience and greenness of the system. We have also designed an experiment protocol and its application to a real CAIS demonstrator. Our approach aims to attack the problem from two perspectives: as a one-agent decision problem through optimization, which takes the decision based on the score of resilience and greenness, and as a two-agent decision problem through game theory, which takes the decision based on the payoff computed for resilience and greenness as two players of a cooperative game.

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 $E'$.

A pair $\langle G_0, G_1 \rangle$ of graphs admits a mutual witness proximity drawing $\langle \Gamma_0, \Gamma_1 \rangle$ when: (i) $\Gamma_i$ represents $G_i$, and (ii) there is an edge $(u,v)$ in $\Gamma_i$ if and only if there is no vertex $w$ in $\Gamma_{1-i}$ that is ``too close'' to both $u$ and $v$ ($i=0,1$). In this paper, we consider infinitely many definitions of closeness by adopting the $\beta$-proximity rule for any $\beta \in [1,\infty]$ and study pairs of isomorphic trees that admit a mutual witness $\beta$-proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness $\beta$-proximity drawing for any $\beta \in [1,\infty]$. The constructive technique can be made ``robust'': For some tree pairs we can suitably prune linearly many leaves from one of the two trees and still retain their mutual witness $\beta$-proximity drawability. Notably, in the special case of isomorphic caterpillars and $\beta=1$, we construct linearly separable mutual witness Gabriel drawings., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

The space sector is experiencing a flourishing growth and evidence is mounting that the near future will be characterized by a large amount of deep-space missions. In the last decade, CubeSats have granted affordable access to space due to their reduced manufacturing costs compared to traditional missions. At the present-day, most miniaturized spacecraft have thus far been deployed into near-Earth orbits, but soon a multitude of interplanetary CubeSats will be employed for deep-space missions as well. Nevertheless, the current paradigm for deep-space missions strongly relies on ground-based operations. Although reliable, this approach will rapidly cause saturation of ground slots, thereby hampering the current momentum in space exploration. At the actual pace, human-in-the-loop, flight-related operations for deep-space missions will soon become unsustainable. Self-driving spacecraft are challenging the current paradigm under which spacecraft are piloted in interplanetary space. They are intended as machines capable of traveling in deep space and autonomously reaching their destination. In EXTREMA, these systems are used to engineer ballistic capture (BC), thereby proving the effectiveness of autonomy in a complex scenario. The key is to accomplish low-thrust orbits culminating in BC. For this, a bundle of BC orbits named ballistic capture corridor (BCC) can be targeted far away from a planet. To achieve BC at Mars without any a priori instruction, an inexpensive and accurate method to construct BCC directly on board is required. Therefore, granting spacecraft the capability to manipulate stable sets in order to self-compute a BCC is crucial. The goal of the paper is to numerically synthesize a corridor exploiting the polynomial chaos expansion (PCE) technique, thereby applying a suited uncertainty propagation technique to BC orbit propagation., Comment: EXTREMA - Engineering Extremely Rare Events in Astrodynamics for Deep-Space Missions in Autonomy, European Research Council (ERC), European Union (EU), Horizon 2020. The content of this document reflects only the author's view. ERC is not responsible for any use that may be made of the information it contains

Upward planarity testing and Rectilinear planarity testing are central problems in graph drawing. It is known that they are both NP-complete, but XP when parameterized by treewidth. In this paper we show that these two problems are W[1]-hard parameterized by treewidth, which answers open problems posed in two earlier papers. The key step in our proof is an analysis of the All-or-Nothing Flow problem, a generalization of which was used as an intermediate step in the NP-completeness proof for both planarity testing problems. We prove that the flow problem is W[1]-hard parameterized by treewidth on planar graphs, and that the existing chain of reductions to the planarity testing problems can be adapted without blowing up the treewidth. Our reductions also show that the known $n^{O(tw)}$-time algorithms cannot be improved to run in time $n^{o(tw)}$ unless ETH fails., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

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., 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)

Predicting the behavior of real-time traffic (e.g., VoIP) in mobility scenarios could help the operators to better plan their network infrastructures and to optimize the allocation of resources. Accordingly, in this work the authors propose a forecasting analysis of crucial QoS/QoE descriptors (some of which neglected in the technical literature) of VoIP traffic in a real mobile environment. The problem is formulated in terms of a multivariate time series analysis. Such a formalization allows to discover and model the temporal relationships among various descriptors and to forecast their behaviors for future periods. Techniques such as Vector Autoregressive models and machine learning (deep-based and tree-based) approaches are employed and compared in terms of performance and time complexity, by reframing the multivariate time series problem into a supervised learning one. Moreover, a series of auxiliary analyses (stationarity, orthogonal impulse responses, etc.) are performed to discover the analytical structure of the time series and to provide deep insights about their relationships. The whole theoretical analysis has an experimental counterpart since a set of trials across a real-world LTE-Advanced environment has been performed to collect, post-process and analyze about 600,000 voice packets, organized per flow and differentiated per codec., Comment: 15 pages

We study the following problem: Given a set $S$ of $n$ points in the plane, how many edge-disjoint plane straight-line spanning paths of $S$ can one draw? A well known result is that when the $n$ points are in convex position, $\lfloor n/2\rfloor$ such paths always exist, but when the points of $S$ are in general position the only known construction gives rise to two edge-disjoint plane straight-line spanning paths. In this paper, we show that for any set $S$ of at least ten points, no three of which are collinear, one can draw at least three edge-disjoint plane straight-line spanning paths of~$S$. Our proof is based on a structural theorem on halving lines of point configurations and a strengthening of the theorem about two spanning paths, which we find interesting in its own right: if $S$ has at least six points, and we prescribe any two points on the boundary of its convex hull, then the set contains two edge-disjoint plane spanning paths starting at the prescribed points., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

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.

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 $1 \leq i,j \leq h$, $i\neq j$.

Orthogonal graph drawings are used in applications such as UML diagrams, VLSI layout, cable plans, and metro maps. We focus on drawing planar graphs and assume that we are given an \emph{orthogonal representation} that describes the desired shape, but not the exact coordinates of a drawing. Our aim is to compute an orthogonal drawing on the grid that has minimum area among all grid drawings that adhere to the given orthogonal representation. This problem is called orthogonal compaction (OC) and is known to be NP-hard, even for orthogonal representations of cycles [Evans et al., 2022]. We investigate the complexity of OC with respect to several parameters. Among others, we show that OC is fixed-parameter tractable with respect to the most natural of these parameters, namely, the number of \emph{kitty corners} of the orthogonal representation: the presence of pairs of kitty corners in an orthogonal representation makes the OC problem hard. Informally speaking, a pair of kitty corners is a pair of reflex corners of a face that point at each other. Accordingly, the number of kitty corners is the number of corners that are involved in some pair of kitty corners., Comment: 40 pages, 26 figures, to appear in Proc. SOFSEM 2023

Let $\Gamma$ be a straight-line drawing of a graph and let $u$ and $v$ be two vertices of $\Gamma$. The Gabriel disk of $u,v$ is the disk having $u$ and $v$ as antipodal points. A pair $\langle \Gamma_0,\Gamma_1 \rangle$ of vertex-disjoint straight-line drawings form a mutual witness Gabriel drawing when, for $i=0,1$, any two vertices $u$ and $v$ of $\Gamma_i$ are adjacent if and only if their Gabriel disk does not contain any vertex of $\Gamma_{1-i}$. We characterize the pairs $\langle G_0,G_1 \rangle $ of complete bipartite graphs that admit a mutual witness Gabriel drawing. The characterization leads to a linear time testing algorithm. We also show that when at least one of the graphs in the pair $\langle G_0, G_1 \rangle $ is complete $k$-partite with $k>2$ and all partition sets in the two graphs have size greater than one, the pair does not admit a mutual witness Gabriel drawing., Comment: Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022)

Motivated by dynamic graph visualization, we study the problem of representing a graph $G$ in the form of a \emph{storyplan}, that is, a sequence of frames with the following properties. Each frame is a planar drawing of the subgraph of $G$ induced by a suitably defined subset of its vertices. Between two consecutive frames, a new vertex appears while some other vertices may disappear, namely those whose incident edges have already been drawn in at least one frame. In a storyplan, each vertex appears and disappears exactly once. For a vertex (edge) visible in a sequence of consecutive frames, the point (curve) representing it does not change throughout the sequence. Note that the order in which the vertices of $G$ appear in the sequence of frames is a total order. In the \textsc{StoryPlan} problem, we are given a graph and we want to decide whether there exists a total order of its vertices 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 $G$. Also, 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 in the case in which the total order of the vertices is given as part of the input and we have to choose how to draw the frames., Comment: Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022)

A graph is rectilinear planar if it admits a planar orthogonal drawing without bends. While testing rectilinear planarity is NP-hard in general (Garg and Tamassia, 2001), it is a long-standing open problem to establish a tight upper bound on its complexity for partial 2-trees, i.e., graphs whose biconnected components are series-parallel. We describe a new O(n^2)-time algorithm to test rectilinear planarity of partial 2-trees, which improves over the current best bound of O(n^3 \log n) (Di Giacomo et al., 2022). Moreover, for partial 2-trees where no two parallel-components in a biconnected component share a pole, we are able to achieve optimal O(n)-time complexity. Our algorithms are based on an extensive study and a deeper understanding of the notion of orthogonal spirality, introduced several years ago (Di Battista et al, 1998) to describe how much an orthogonal drawing of a subgraph is rolled-up in an orthogonal drawing of the graph., Comment: arXiv admin note: substantial text overlap with arXiv:2110.00548 Appears in the Proceedings of the 30th International Symposium on Graph Drawing and Network Visualization (GD 2022)

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 sequence of horizontal and vertical segments between its end-points. 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., Comment: arXiv admin note: text overlap with arXiv:2008.03784

Affective computing plays a key role in human-computer interactions, entertainment, teaching, safe driving, and multimedia integration. Major breakthroughs have been made recently in the areas of affective computing (i.e., emotion recognition and sentiment analysis). Affective computing is realized based on unimodal or multimodal data, primarily consisting of physical information (e.g., textual, audio, and visual data) and physiological signals (e.g., EEG and ECG signals). Physical-based affect recognition caters to more researchers due to multiple public databases. However, it is hard to reveal one's inner emotion hidden purposely from facial expressions, audio tones, body gestures, etc. Physiological signals can generate more precise and reliable emotional results; yet, the difficulty in acquiring physiological signals also hinders their practical application. Thus, the fusion of physical information and physiological signals can provide useful features of emotional states and lead to higher accuracy. Instead of focusing on one specific field of affective analysis, we systematically review recent advances in the affective computing, and taxonomize unimodal affect recognition as well as multimodal affective analysis. Firstly, we introduce two typical emotion models followed by commonly used databases for affective computing. Next, we survey and taxonomize state-of-the-art unimodal affect recognition and multimodal affective analysis in terms of their detailed architectures and performances. Finally, we discuss some important aspects on affective computing and their applications and conclude this review with an indication of the most promising future directions, such as the establishment of baseline dataset, fusion strategies for multimodal affective analysis, and unsupervised learning models., Comment: Accepted for Information Fusion

Using a unique dataset containing gridded data on population densities, rents, housing sizes, and transportation in 192 cities worldwide, we investigate the empirical relevance of the monocentric standard urban model (SUM). Overall, the SUM seems surprisingly capable of capturing the inner structure of cities, both in developed and developing countries. As expected, cities spread out when they are richer, more populated, and when transportation or farmland is cheaper. Respectively 100% and 87% of the cities exhibit the expected negative density and rent gradients: on average, a 1% decrease in income net of transportation costs leads to a 21% decrease in densities and a 3% decrease in rents per m2. We also investigate the heterogeneity between cities of different characteristics in terms of monocentricity, informality, and amenities.

We study the long-standing open problem of efficiently testing rectilinear planarity of series-parallel graphs (SP-graphs) in the variable embedding setting. A key ingredient behind the design of a linear-time testing algorithm for SP-graphs of vertex-degree at most three is that one can restrict the attention to a constant number of ``rectilinear shapes'' for each series or parallel component. To formally describe these shapes the notion of spirality can be used. This key ingredient no longer holds for SP-graphs with vertices of degree four, as we prove a logarithmic lower bound on the spirality of their components. The bound holds even for the independent-parallel SP-graphs, in which no two parallel components share a pole. Nonetheless, by studying the spirality properties of the independent-parallel SP-graphs, we are able to design a linear-time rectilinear planarity testing algorithm for this graph family.