arXiv subjects

Fabrizio Grosso

Publications and source records attributed to Fabrizio Grosso.

10 recordsLinked to original sources

arXiv2026

Extending Biconnected Straight-Line Planar Drawings

The Partial Drawing Extensibility problem, for short PDE, takes as input a triple $\langle G,H,\Gamma_H\rangle$, where $G$ is a planar graph, $H$ is a subgraph of $G$, and $\Gamma_H$ is a straight-line planar drawing of $H$, and asks whether $\Gamma_H$ can be extended to a straight-line planar drawing of $G$. Patrignani [Int. J. Found. Comput. Sci. (2006)] proved that the PDE problem is NP-hard, exploiting instances in which $H$ is highly disconnected. In this paper, we study the PDE problem under the requirement that the initial partial drawing $\Gamma_H$ is biconnected. We show that PDE remains NP-hard even for instances in which $H$ is a biconnected graph with faces of bounded size, $G$ is subcubic, and the part of $G$ that is not in $H$ consists of length-$2$ paths. The complexity of PDE remains however open when $H$ is connected (or even biconnected) if $G$ has a fixed embedding. In this setting both a polynomial-time algorithm or an NP-hardness proof seem to be elusive targets. As a step towards tackling this problem, we study instances of PDE in which $H$ is biconnected, $G$ has a fixed embedding, and the rest of the graph consists of $p$ length-2 paths, and present an $O(p^2 n)$-time algorithm, a result in sharp contrast with the NP-hardness of the variable embedding setting. Moreover, with an approach based on the Existential Theory of the Reals, we show that, if $H$ is biconnected, the problem is FPT parameterized by the vertex cover number of $G$, both in a fixed and in a variable embedding setting.

cs.CG↗

arXiv2026

Design Space and Implementation of RAG-Based Avatars for Virtual Archaeology

Immersive technologies, such as virtual and augmented reality, are transforming digital heritage by enabling users to explore and interact with culturally significant sites. It is now possible to view and augment digital twins, or digitally reconstructed versions of them, and to enable access to previously unreachable locations for a broader audience. Here, we investigate retrieval-augmented generation (RAG)-based avatars as an interface for accessing further information about digital cultural heritage objects while immersed in dedicated virtual environments. We present a requirement design space that spans the application realm, avatar personality, and I/O modalities. We instantiate it with a RAG system coupled to a conversational avatar in a virtual reality (VR) environment, using the Maxentius mausoleum from the 4th century AD as a case study, through which users gain access to curated on-demand information of the digitised heritage object. Our workflow utilises scholarly texts and enriches them with metadata. We evaluate various RAG configurations in terms of answer quality on a small expert-crafted question-answer set, as well as the perceived workload of users of a VR setup using such a RAG avatar. We demonstrate evidence that users perceive the overall workload for interacting with such an avatar as below average and that such avatars help to gain topical engagement. Overall, our work demonstrates how to utilise RAG-driven VR avatars for archaeological purposes and provides evidence that they can offer a pathway for immersive, AI-enhanced digital heritage applications.

cs.HC↗

arXiv2025

A Walk on the Wild Side: a Shape-First Methodology for Orthogonal Drawings

Several algorithms for the construction of orthogonal drawings of graphs, including those based on the Topology-Shape-Metrics (TSM) paradigm, tend to prioritize the minimization of crossings. This emphasis has two notable side effects: some edges are drawn with unnecessarily long sequences of segments and bends, and the overall drawing area may become excessively large. As a result, the produced drawings often lack geometric uniformity. Moreover, orthogonal crossings are known to have a limited impact on readability, suggesting that crossing minimization may not always be the optimal goal. In this paper, we introduce a methodology that 'subverts' the traditional TSM pipeline by focusing on minimizing bends. Given a graph $G$, we ideally seek to construct a rectilinear drawing of $G$, that is, an orthogonal drawing with no bends. When not possible, we incrementally subdivide the edges of $G$ by introducing dummy vertices that will (possibly) correspond to bends in the final drawing. This process continues until a rectilinear drawing of a subdivision of the graph is found, after which the final coordinates are computed. We tackle the (NP-complete) rectilinear drawability problem by encoding it as a SAT formula and solving it with state-of-the-art SAT solvers. If the SAT formula is unsatisfiable, we use the solver's proof to determine which edge to subdivide. Our implementation, DOMUS, which is fairly simple, is evaluated through extensive experiments on small- to medium-sized graphs. The results show that it consistently outperforms OGDF's TSM-based approach across most standard graph drawing metrics.

cs.CG↗

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↗

arXiv2025

Upward Pointset Embeddings of Planar st-Graphs

We study upward pointset embeddings (UPSEs) of planar $st$-graphs. Let $G$ be a planar $st$-graph and let $S \subset \mathbb{R}^2$ be a pointset with $|S|= |V(G)|$. An UPSE of $G$ on $S$ is an upward planar straight-line drawing of $G$ that maps the vertices of $G$ to the points of $S$. We consider both the problem of testing the existence of an UPSE of $G$ on $S$ (UPSE Testing) and the problem of enumerating all UPSEs of $G$ on $S$. We prove that UPSE Testing is NP-complete even for $st$-graphs that consist of a set of directed $st$-paths sharing only $s$ and $t$. On the other hand, if $G$ is an $n$-vertex planar $st$-graph whose maximum $st$-cutset has size $k$, then UPSE Testing can be solved in $O(n^{4k})$ time with $O(n^{3k})$ space; also, all the UPSEs of $G$ on $S$ can be enumerated with $O(n)$ worst-case delay, using $O(k n^{4k} \log n)$ space, after $O(k n^{4k} \log n)$ set-up time. Moreover, for an $n$-vertex $st$-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given pointset, which can be tested in $O(n \log n)$ time. Related to this result, we give an algorithm that, for a set $S$ of $n$ points, enumerates all the non-crossing monotone Hamiltonian cycles on $S$ with $O(n)$ worst-case delay, using $O(n^2)$ space, after $O(n^2)$ set-up time.

cs.DS↗

arXiv2024

Ranking and Unranking of the Planar Embeddings of a Planar Graph

Let $\mathcal{G}$ be the set of all the planar embeddings of a (not necessarily connected) $n$-vertex graph $G$. We present a bijection $Φ$ from $\mathcal{G}$ to the natural numbers in the interval $[0 \dots |\mathcal{G}| - 1]$. Given a planar embedding $\mathcal{E}$ of $G$, we show that $Φ(\mathcal{E})$ can be decomposed into a sequence of $O(n)$ natural numbers each describing a specific feature of $\mathcal{E}$. The function $Φ$, which is a ranking function for $\mathcal{G}$, can be computed in $O(n)$ time, while its inverse unranking function $Φ^{-1}$ can be computed in $O(n α(n))$ time. The results of this paper can be of practical use to uniformly at random generating the planar embeddings of a graph $G$ or to enumerating such embeddings with amortized constant delay. Also, they can be used to counting, enumerating or uniformly at random generating constrained planar embeddings of $G$.

cs.CG↗

arXiv2024

Unit-length Rectangular Drawings of Graphs

A rectangular drawing of a planar graph $G$ is a planar drawing of $G$ in which vertices are mapped to grid points, edges are mapped to horizontal and vertical straight-line segments, and faces are drawn as rectangles. Sometimes this latter constraint is relaxed for the outer face. In this paper, we study rectangular drawings in which the edges have unit length. We show a complexity dichotomy for the problem of deciding the existence of a unit-length rectangular drawing, depending on whether the outer face must also be drawn as a rectangle or not. Specifically, we prove that the problem is NP-complete for biconnected graphs when the drawing of the outer face is not required to be a rectangle, even if the sought drawing must respect a given planar embedding, whereas it is polynomial-time solvable, both in the fixed and the variable embedding settings, if the outer face is required to be drawn as a rectangle.

cs.CG↗

arXiv2023

Efficient Enumeration of Drawings and Combinatorial Structures for Maximal Planar Graphs

We propose efficient algorithms for enumerating the notorious combinatorial structures of maximal planar graphs, called canonical orderings and Schnyder woods, and the related classical graph drawings by de Fraysseix, Pach, and Pollack [Combinatorica, 1990] and by Schnyder [SODA, 1990], called canonical drawings and Schnyder drawings, respectively. To this aim (i) we devise an algorithm for enumerating special $e$-bipolar orientations of maximal planar graphs, called canonical orientations; (ii) we establish bijections between canonical orientations and canonical drawings, and between canonical orientations and Schnyder drawings; and (iii) we exploit the known correspondence between canonical orientations and canonical orderings, and the known bijection between canonical orientations and Schnyder woods. All our enumeration algorithms have $O(n)$ setup time, space usage, and delay between any two consecutively listed outputs, for an $n$-vertex maximal planar graph.

cs.DS↗

arXiv2023

Treebar Maps: Schematic Representation of Networks at Scale

Many data sets, crucial for today's applications, consist essentially of enormous networks, containing millions or even billions of elements. Having the possibility of visualizing such networks is of paramount importance. We propose an algorithmic framework and a visual metaphor, dubbed treebar map, to provide schematic representations of huge networks. Our goal is to convey the main features of the network's inner structure in a straightforward, two-dimensional, one-page drawing. This drawing effectively captures the essential quantitative information about the network's main components. Our experiments show that we are able to create such representations in a few hundreds of seconds. We demonstrate the metaphor's efficacy through visual examination of extensive graphs, highlighting how their diverse structures are instantly comprehensible via their representations.

cs.DS↗

arXiv2022

Small Point-Sets Supporting Graph Stories

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

cs.DS↗