arXiv subjects

Emilio Di Giacomo

Publications and source records attributed to Emilio Di Giacomo.

At least 19 recordsLinked to original sources

arXiv2026

Stack and Queue Layouts with Defects

Linear layouts of graphs -- particularly \emph{stack} and \emph{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 \emph{stack number} or \emph{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 \emph{$k$-defective stack layout} (resp. a \emph{$k$-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.

cs.CG↗

arXiv2026

Minimum Monotone Spanning Trees

Given a finite set $S$ of points in the plane and a finite set $\mathcal{D}$ of directions, a geometric spanning tree~$T$ of~$S$ is $\mathcal{D}$-monotone if every path in $T$ is monotone with respect to some direction in $\mathcal{D}$. We study the problem of computing, for a given point set $S$ and a given set $\mathcal{D}$ of directions, a minimum-length $\mathcal{D}$-monotone spanning tree of~$S$. We present a quadratic-time algorithm for two directions. More generally, we show that the problem belongs to the complexity class XP when parameterized by the number of directions. We further study, for a given positive integer $k$ and point set~$S$, the problem of finding a minimum-length $\mathcal{D}$-monotone spanning tree of $S$ over all possible sets~$\mathcal{D}$ of $k$ directions. We prove that this problem, too, is in XP when parameterized by~$k$, and present two algorithms that run in $O(n^2 \log n)$ and $O(n^6)$ time for $k=1$ and $k=2$, respectively, where $n$ is the number of points in~$S$. Finally, in contrast to the classical Euclidean minimum spanning tree of a set of points, whose vertex degree is bounded by six, we show that for every even integer~$k$, there exists a point set~$S_k$ and a set $\mathcal{D}_k$ of $k$ directions such that any minimum-length $\mathcal{D}_k$-monotone spanning tree of $S_k$ has maximum vertex degree~$2k$.

cs.CG↗

arXiv2026

Collaborative Problem Solving in Mixed Reality: A Study on Visual Graph Analysis

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.

cs.HC↗

arXiv2025

Linear Layouts of Graphs with Priority Queues

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.

cs.DM↗

arXiv2025

Unit Edge-Length Rectilinear Drawings with Crossings and Rectangular Faces

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.

cs.CG↗

arXiv2024

On 1-bend Upward Point-set Embeddings of $st$-digraphs

We study the upward point-set embeddability of digraphs on one-sided convex point sets with at most 1 bend per edge. We provide an algorithm to compute a 1-bend upward point-set embedding of outerplanar $st$-digraphs on arbitrary one-sided convex point sets. We complement this result by proving that for every $n \geq 18$ there exists a $2$-outerplanar $st$-digraph $G$ with $n$ vertices and a one-sided convex point set $S$ so that $G$ does not admit a 1-bend upward point-set embedding on $S$.

cs.CG↗

arXiv2023

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

The $\textit{planar slope number}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G) \in O(c^Δ)$ for every planar graph $G$ of maximum degree $Δ$. This upper bound has been improved to $O(Δ^5)$ if $G$ has treewidth three, and to $O(Δ)$ if $G$ has treewidth two. In this paper we prove $psn(G) \leq \max\{4,Δ\}$ 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 $O(Δ^2)$ slopes suffice for nested pseudotrees.

cs.CG↗

arXiv2023

New Bounds on the Local and Global Edge-length Ratio of Planar Graphs

The \emph{local edge-length ratio} of a planar straight-line drawing $Γ$ is the largest ratio between the lengths of any pair of edges of $Γ$ that share a common vertex. The \emph{global edge-length ratio} of $Γ$ is the largest ratio between the lengths of any pair of edges of $Γ$. 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 $Ω(\sqrt{n})$. 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.

cs.CG↗

arXiv2023

On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity

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.

cs.CG↗

arXiv2023

$k$-planar Placement and Packing of $Δ$-regular Caterpillars

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 $Δ$-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree $Δ$. 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 $h$ $Δ$-regular caterpillars and sufficient conditions for the packing of a set of $Δ_1$-, $Δ_2$-, $\dots$, $Δ_h$-regular caterpillars such that the degree $Δ_i$ and the degree $Δ_j$ of the non-leaf vertices can differ from one caterpillar to another, for $1 \leq i,j \leq h$, $i\neq j$.

math.CO↗

arXiv2022

On the Complexity of the Storyplan Problem

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.

cs.CC↗

arXiv2022

Testing Upward Planarity of Partial $2$-Trees

We present an $O(n^2)$-time algorithm to test whether an $n$-vertex directed partial $2$-tree is upward planar. This result improves upon the previously best known algorithm, which runs in $O(n^4)$ time.

cs.DS↗

arXiv2022

Parameterized Algorithms for Upward Planarity

We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, single-source digraphs and outerplanar digraphs.

cs.CG↗

arXiv2021

Quasi-upward Planar Drawings with Minimum Curve Complexity

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.

cs.CG↗

arXiv2021

A User Study on Hybrid Graph Visualizations

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.

cs.HC↗

arXiv2020

Storyline Visualizations with Ubiquitous Actors

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.

cs.SI↗

arXiv2019

Packing Trees into 1-planar Graphs

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 $n \leq 10$.

cs.CG↗

arXiv2019

Sketched Representations and Orthogonal Planarity of Bounded Treewidth Graphs

Given a planar graph $G$ and an integer $b$, OrthogonalPlanarity is the problem of deciding whether $G$ admits an orthogonal drawing with at most $b$ bends in total. We show that OrthogonalPlanarity can be solved in polynomial time if $G$ has bounded treewidth. Our proof is based on an FPT algorithm whose parameters are the number of bends, the treewidth and the number of degree-2 vertices of $G$. This result is based on the concept of sketched orthogonal representation that synthetically describes a family of equivalent orthogonal representations. Our approach can be extended to related problems such as HV-Planarity and FlexDraw. In particular, both OrthogonalPlanarity and HV-Planarity can be decided in $O(n^3 \log n)$ time for series-parallel graphs, which improves over the previously known $O(n^4)$ bounds.

cs.CG↗