# Emilio Di GiacomoUniversità degli Studi di Perugia | UNIPG · Department of Electronic and Information Engineering

Emilio Di Giacomo

PhD Information Engineering

## About

136

Publications

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## Publications

Publications (136)

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 wh...

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.

We prove that every set $\mathcal S$ of $\Delta$ slopes containing the horizontal slope is universal for $1$-bend upward planar drawings of bitonic $st$-graphs with maximum vertex degree $\Delta$, i.e., every such digraph admits a $1$-bend upward planar drawing whose edge segments use only slopes in $\mathcal S$. This result is worst-case optimal i...

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 thre...

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 seri...

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 s...

Given a graph G and an integer b, OrthogonalPlanarity is the problem of testing whether G admits a planar orthogonal drawing with at most b bends. OrthogonalPlanarity is known to be NP-complete. We show that this problem belongs to the XP class when parameterized by treewidth. The proof exploits a fixed-parameter tractable approach that uses two mo...

We present a new model for hybrid planarity that relaxes existing hybrid representation models. 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 di...

Let G be a planar graph whose vertices are colored either red or blue and let S be a set of points having as many red (resp. blue) points as the red (resp. blue) vertices of G. A 2-colored point-set embedding of G on S is a planar drawing that maps each red (resp. blue) vertex of G to a red (resp. blue) point of S. We show that there exist partial...

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 s...

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 seri...

The planar slope number \(\mathrm {psn}(G)\) of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that \(\mathrm {psn}(G) \in O(c^{\varDelta })\) for every planar graph G of degree \(\varDelta \). This upper bound has been improved to \(O(\varDelta ^5)\) if G has treewidth three, and to \(O(\v...

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 $psn(G) \in O(c^\Delta)$ for every planar graph $G$ of degree $\Delta$. This upper bound has been improved to $O(\Delta^5)$ if $G$ has treewidth three, and to $O(\Delta)$ if $G$ has treewi...

Let G be a planar graph whose vertices are colored either red or blue and let S be a set of points having as many red (blue) points as the red (blue) vertices of G. A 2-colored point-set embedding of G on S is a planar drawing that maps each red (blue) vertex of G to a red (blue) point of S. We show that there exist properly 2-colored graphs (i.e.,...

This paper describes the research activity on financial crime detection developed by the computer engineering group at the University of Perugia. The presented research aims at designing and experimenting advanced visual interfaces to support financial crime detection, with a focus on tax evasion discovery. The activity of the group on this topic,...

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 appli...

One of the primary goals of many systems for the visual analysis of dynamically changing networks is to maintain the stability of the drawing throughout the sequence of graph changes. We investigate the scenario where the changes are determined by a stream of events, each being either an edge addition or an edge removal. The visualization must be u...

We establish new results on the curve complexity of k-colored point-set embeddings when k=3. We show that there exist 3-colored caterpillars with only three independent edges whose 3-colored point-set embeddings may require Ω(n13) bends on Ω(n23) edges. This settles an open problem by Badent et al. [5] about the curve complexity of point set embedd...

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 appli...

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...

It is proved that every series-parallel digraph whose maximum vertex degree is Δ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Δ distinct slopes. The construction is worst-case optimal in terms of the number of slopes, and it gives rise to drawings with optimal angular resolution πΔ. A variant...

We study the problem of representing topological graphs as polyline drawings with few bends per edge and such that the topology of the graph is either fully or partially preserved. More formally, let G be a simple topological graph and let Γ be a polyline drawing of G. Drawing Γ partially preserves the topology of G if it has the same external boun...

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...

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 pat...

In a visibility representation of a graph G, the vertices are represented by non-overlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility repres...

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...

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 tre...

A topological graph is a graph drawn in the plane. A topological graph is k-plane, k>0, if each edge is crossed at most k times. We study the problem of partitioning the edges of a k-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for k=1, we focus on optimal 2-plane and on optimal...

We study $k$-page upward book embeddings ($k$UBEs) of $st$-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on $k$ pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a $k$UBE is NP-complete...

We prove that every set \(\mathcal {S}\) of \(\varDelta \) slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree \(\varDelta \), i.e., every such digraph admits a 1-bend upward planar drawing whose edge segments use only slopes in \(\mathcal {S}\). This result is worst-...

Let $G$ be a simple topological graph and let $\Gamma$ be a polyline drawing of $G$. We say that $\Gamma$ \emph{partially preserves the topology} of $G$ if it has the same external boundary, the same rotation system, and the same set of crossings as $G$. Drawing $\Gamma$ fully preserves the topology of $G$ if the planarization of $G$ and the planar...

A topological graph is a graph drawn in the plane. A topological graph is k-plane, \(k>0\), if each edge is crossed at most k times. We study the problem of partitioning the edges of a k-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for \(k=1\), we focus on optimal 2-plane and 3-p...

A graph $G$ is $(k,p)$-planar if its vertices 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 region in the plane (called cluster region) that is topologically equivalent to a disk; (ii) Cluster regions are pairwise disjoint; (iii) Each vertex of a cluster is as...

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal $2$-plane and...

A subcubic planar graph is a planar graph whose vertices have degree at most 3. We show that the subcubic planar graphs with at least five vertices have planar slope number at most 4, which is worst-case optimal. This answers an open question by Jelínek et al. Furthermore, we prove that the subcubic planar graphs with at least five vertices have an...

We show that any planar drawing of a forest of three stars whose vertices are constrained to be at fixed vertex locations may require $\Omega(n^\frac{2}{3})$ edges each having $\Omega(n^\frac{1}{3})$ bends in the worst case. The lower bound holds even when the function that maps vertices to points is not a bijection but it is defined by a 3-colorin...

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 arbitrarily chosen. We show that NodeTrix planarity testing with fixed sides ca...

A drawing of a graph such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line is called a 2-page drawing. If all edges are in the same half-plane, the drawing is called a 1-page drawing. We want to compute 1-page and 2-page drawings of planar graphs such that the numb...

A $1$-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a $1$-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a $1$-plane graph $G$ is a coloring of the edges of $G$ with two colors, red and blue, such that both the graph induced by...

Content-based travel recommender systems suggest touristic attractions based on a best match between users’ preferences and a given set of points of interests, called POIs for short. When designing such systems, a critical aspect is to equip them with a rich enough knowledge base that, for each POI, indicates how much the POI is relevant for a set...

The ply number of a drawing is a new criterion of interest for graph drawing. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the longest edge incident to that vertex. This paper reports the...

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at mo...

It is proved that every series-parallel digraph whose maximum vertex-degree is \(\varDelta \) admits an upward planar drawing with at most one bend per edge such that each edge segment has one of \(\varDelta \) distinct slopes. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawi...

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane \(z=0\) and edges are unobstructed lines of sight parallel to the x- or y-axis. We prove that: (i) Every complete bipartite graph admits a 2.5D-BR; (ii) The complete graph \(K_n\) admits a 2.5D-BR if and o...

We study the problem of computing drawings of planar graphs in sub-quadratic area, by allowing edge crossings. We first prove that sub-quadratic area cannot be achieved if only a constant number of crossings per edge is allowed. More precisely, we show that the same area lower bounds as in the crossing-free case hold for straight-line and poly-line...

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if a...

It is proved that every series-parallel digraph whose maximum vertex-degree is $\Delta$ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of $\Delta$ distinct slopes. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawings with opt...

Let G be a 3-connected planar graph with n vertices and let be the maximum number of vertices of an induced subgraph of G that is a path. Substantially improving previous results, we prove that . To demonstrate the tightness of this bound, we notice that the above inequality implies , where ε is any positive constant smaller than 1, and describe an...

An ortho-polygon visibility representation of an $n$-vertex embedded graph $G$ (OPVR of $G$) is an embedding preserving drawing of $G$ that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of $G$ is the minimum $k$ such that every polyg...

A 2-page drawing of a graph is such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line. If all edges are in the same half-plane, the drawing is called a 1-page drawing. We want to compute 1-page and 2-page drawings of planar graphs such that the number of crossings p...

A simultaneous geometric embedding (SGE) of two planar graphs $G_1$ and $G_2$ with the same vertex set is a pair of straight-line planar drawings $\Gamma _1$ of $G_1$ and $\Gamma _2$ of $G_2$ such that each vertex is drawn at the same point in $\Gamma _1$ and $\Gamma _2$. Many papers have been devoted to the study of which pairs of graphs admit a S...

We consider the problem of characterizing graphs with low ply number and algorithms for creating layouts of graphs with low ply number. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the lo...

Touristic data represent a valuable and useful source of knowledge for local administrators. Using visualization systems helps them in understanding the main trends of touristic flows directed to the different attractions of their territory and thus it supports their decision making processes. In fact, the problem of visualizing flows of people (or...

The impressive and pervasive growth of different mobile devices allows people to quickly access and share a large variety of multimedia contents. These devices offer a heterogeneous set of display capabilities, with several types of resolution levels and screen sizes. This scenario often requires to adapt the displayed images to the aspect ratio of...

A graph is outer 1-planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1-planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3-trees. We show that an outer 1-planar graph G of bounded degree △ admits an outer 1-planar straight-line dra...

A simultaneous geometric embedding (SGE) of two planar graphs G 1 and G 2 with the same vertex set is a pair of straight-line planar drawings Γ1 of G 1 and Γ2 of G 2 such that each vertex is drawn at the same point in Γ1 and Γ2. Many papers have been devoted to the study of which pairs of graphs admit a SGE, and both positive and negative results h...

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every n-vertex fan-planar drawing has at most 5n − 10 edges, and that this bound is tight for n ≥ 20. We extend their result from both the combinatorial and the algorit...

We propose an approach that allows a user (e.g., an analyst) to explore a layout produced by any graph drawing algorithm, in order to reduce the visual complexity and clarify its presentation. Our approach is based on stratifying the drawing into layers with desired properties; to this aim, heuristics are presented. The produced layers can be explo...

Upward drawing is a widely studied drawing convention for the visual representation of directed graphs. In an upward drawing vertices are mapped to distinct points of the plane, and edges are curves monotonically increasing in the vertical direction, according to their orientation. In particular, not all planar digraphs admit an upward planar drawi...

In a \emph{fan-planar drawing} of a graph an edge can cross only edges with a
common end-vertex. Fan-planar drawings have been recently introduced by
Kaufmann and Ueckerdt, who proved that every $n$-vertex fan-planar drawing has
at most $5n-10$ edges, and that this bound is tight for $n \geq 20$. We extend
their result, both from the combinatorial...

A subcubic planar graph is a planar graph whose vertices have degree at most three. We show that the subcubic planar graphs with at least five vertices have planar slope number at most four, which is worst case optimal. This answers an open question by Jelínek et al. [6]. As a corollary, we prove that the subcubic planar graphs with at least five v...

We propose an approach that allows a user (e.g., an analyst) to explore a layout produced by any graph drawing algorithm, in order to reduce the visual complexity and clarify its presentation. Our approach is based on stratifying the drawing into layers with desired properties; to this aim, heuristics are presented. The produced layers can be explo...

In this paper we study how to compute compact straight-line drawings of planar graphs with a limited number of crossings per edge. We prove that every outerplanar graph can be drawn in O(nlogn) area using a sub-linear number of crossings per edge, and that for any given number ε>0ε>0, every outerplanar graph admits an O(n1+ε)O(n1+ε) area drawing wi...

Let G be a 3-connected planar graph with n vertices and let p(G) be the maximum number of vertices of an induced subgraph of G that is a path. We prove that \(p(G) \geq \frac{\log n}{12 \log \log n}\). To demonstrate the tightness of this bound, we notice that the above inequality implies p(G) ∈ Ω((log2
n)1 − ε
), where ε is any positive constant s...

Questo capitolo presenta due studi: il primo riguarda la progettazione e il processo di valutazione di uno strumento finalizzato a estendere agli utenti disabili la possibilità di ricercare e accedere alle informazioni su Internet (WhatsOnWeb); il secondo è dedicato allo sviluppo di uno strumento di telemedicina per la riabilitazione (Nu!Reha). Wha...

It is proven that every set $S$ of distinct points in the plane with
cardinality $\lceil \frac{\sqrt{\log_2 n}-1}{4} \rceil$ can be a subset of the
vertices of a crossing-free straight-line drawing of any planar graph with $n$
vertices. It is also proven that if $S$ is restricted to be a one-sided convex
point set, its cardinality increases to $\lc...

We study the problem of computing h-quasi planar drawings in linear area; in an h-quasi planar drawing the number of mutually crossing edges is at most h−1. We prove that every n-vertex partial k-tree admits a straight-line h-quasi planar drawing in O(n) area, where h depends on k but not on n. For specific sub-families of partial k-trees, we prese...