arXiv subjects

Giuseppe Di Battista

Publications and source records attributed to Giuseppe Di Battista.

At least 19 recordsLinked to original sources

arXiv2026

Faster Planar Graph Algorithms for Connectivity Problems via Meanders

In this paper, we refine the dynamic programming framework based on the sphere cut decomposition designed by Dorn, Penninkx, Bodlaender, and Fomin (ESA 2005) to obtain faster subexponential algorithms for connectivity problems on planar graphs. We investigate the relationship between these problems and meanders, which are simple closed planar loops that intersect a fixed line in a given number of points. By combining dynamic programming with techniques from meandric system analysis and the use of fast matrix multiplication by Dorn (ESA 2006), we obtain improved algorithms for planar connectivity problems. We show that the number of meanders on $2n$ crossings $M_n$ is $\mathcal O^*(12.806^n)$, which improves the previous upper bound of $\mathcal O^*(12.901^n)$ by Albert and Paterson (FPSAC 2004). This then gives the best-known classical upper bounds on the deterministic time complexity of several planar graph problems with polynomially-bounded weights, namely $\mathcal O(2^{5.543\sqrt n})$ for the Planar Travelling Salesman problem, $\mathcal O(2^{5.796\sqrt n})$ for Planar Longest Cycle/Path, $\mathcal O(2^{8.251\sqrt n})$ for Planar Connected Dominating Set and $\mathcal O(2^{8.037\sqrt n})$ for Planar Steiner Tree. Notably, this leads to the best-known deterministic complexity $\mathcal O(2^{5.543\sqrt{n}})$ for the Planar Hamiltonian Cycle problem.

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arXiv2026

Extending Biconnected Straight-Line Planar Drawings

The Partial Drawing Extensibility problem, for short PDE, takes as input a triple $\langle G,H,Γ_H\rangle$, where $G$ is a planar graph, $H$ is a subgraph of $G$, and $Γ_H$ is a straight-line planar drawing of $H$, and asks whether $Γ_H$ 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 $Γ_H$ 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.

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arXiv2025

On Planar Straight-Line Dominance Drawings

We study the following question, which has been considered since the 90's: Does every $st$-planar graph admit a planar straight-line dominance drawing? We show concrete evidence for the difficulty of this question, by proving that, unlike upward planar straight-line drawings, planar straight-line dominance drawings with prescribed $y$-coordinates do not always exist and planar straight-line dominance drawings cannot always be constructed via a contract-draw-expand inductive approach. We also show several classes of $st$-planar graphs that always admit a planar straight-line dominance drawing. These include $st$-planar $3$-trees in which every stacking operation introduces two edges incoming into the new vertex, $st$-planar graphs in which every vertex is adjacent to the sink, $st$-planar graphs in which no face has the left boundary that is a single edge, and $st$-planar graphs that have a leveling with span at most two.

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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.

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arXiv2025

Tangling and Untangling Trees on Point-sets

We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set $S$ of points, while ensuring that its curve complexity (i.e., maximum number of bends per edge) is bounded by a constant. We focus on trees: Let $T$ be a tree, $\vartheta(T)$ be its thrackle number, and $χ$ be any integer in the interval $[0,\vartheta(T)]$. In the tangling phase we compute a topological linear embedding of $T$ with $\vartheta(T)$ edge crossings and a constant number of spine traversals. In the untangling phase we remove edge crossings without increasing the spine traversals until we reach $χ$ crossings. The computed linear embedding is used to construct a drawing of $T$ on $S$ with $χ$ crossings and constant curve complexity. Our approach gives rise to an $O(n^2)$-time algorithm for general trees and an $O(n \log n)$-time algorithm for paths. We also adapt the approach to compute RAC drawings, i.e. drawings where the angles formed at edge crossings are $\fracπ{2}$.

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arXiv2025

Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem $\mathcal P$ and an instance $I$ of $\mathcal P$, such a speedup can be expressed in terms of the average degree $δ$ of the dependency digraph $G_{\mathcal{P}}(I)$ of $I$, determined by a recursive formulation of $\mathcal P$. The nodes of this graph are the subproblems of $\mathcal P$ induced by $I$ and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in $\tilde{O}(|V(G_{\mathcal{P}}(I))| \sqrtδ)$ time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted $n$-vertex digraph with $m$ edges that runs in $\tilde{O}(n\sqrt{nm})$ time, which improves the best known classical upper bound when $m \in Ω(n^{1.4})$.

quant-ph↗

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.

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arXiv2025

From Tutte to Floater and Gotsman: On the Resolution of Planar Straight-line Drawings and Morphs

The algorithm of Tutte for constructing convex planar straight-line drawings and the algorithm of Floater and Gotsman for constructing planar straight-line morphs are among the most popular graph drawing algorithms. In this paper, focusing on maximal plane graphs, we prove upper and lower bounds on the resolution of the planar straight-line drawings produced by Floater's algorithm, which is a broad generalization of Tutte's algorithm. Further, we use such results in order to prove a lower bound on the resolution of the drawings of maximal plane graphs produced by Floater and Gotsman's morphing algorithm. Finally, we show that such a morphing algorithm might produce drawings with exponentially-small resolution, even when transforming drawings with polynomial resolution.

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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.

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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 $G$.

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arXiv2024

Quantum Algorithms for One-Sided Crossing Minimization

We present singly-exponential quantum algorithms for the One-Sided Crossing Minimization (OSCM) problem. Given an $n$-vertex bipartite graph $G=(U,V,E\subseteq U \times V)$, a $2$-level drawing $(π_U,π_V)$ of $G$ is described by a linear ordering $π_U: U \leftrightarrow \{1,\dots,|U|\}$ of $U$ and linear ordering $π_V: V \leftrightarrow \{1,\dots,|V|\}$ of $V$. For a fixed linear ordering $π_U$ of $U$, the OSCM problem seeks to find a linear ordering $π_V$ of $V$ that yields a $2$-level drawing $(π_U,π_V)$ of $G$ with the minimum number of edge crossings. We show that OSCM can be viewed as a set problem over $V$ amenable for exact algorithms with a quantum speedup with respect to their classical counterparts. First, we exploit the quantum dynamic programming framework of Ambainis et al. [Quantum Speedups for Exponential-Time Dynamic Programming Algorithms. SODA 2019] to devise a QRAM-based algorithm that solves OSCM in $O^*(1.728^n)$ time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in $O^*(2^n)$ time and polynomial space.

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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.

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arXiv2024

Min-$k$-planar Drawings of Graphs

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.

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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.

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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.

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arXiv2023

Quantum Graph Drawing

In this paper, we initiate the study of quantum algorithms in the Graph Drawing research area. We focus on two foundational drawing standards: 2-level drawings and book layouts. Concerning $2$-level drawings, we consider the problems of obtaining drawings with the minimum number of crossings, $k$-planar drawings, quasi-planar drawings, and the problem of removing the minimum number of edges to obtain a $2$-level planar graph. Concerning book layouts, we consider the problems of obtaining $1$-page book layouts with the minimum number of crossings, book embeddings with the minimum number of pages, and the problem of removing the minimum number of edges to obtain an outerplanar graph. We explore both the quantum circuit and the quantum annealing models of computation. In the quantum circuit model, we provide an algorithmic framework based on Grover's quantum search, which allows us to obtain, at least, a quadratic speedup on the best classical exact algorithms for all the considered problems. In the quantum annealing model, we perform experiments on the quantum processing unit provided by D-Wave, focusing on the classical $2$-level crossing minimization problem, demonstrating that quantum annealing is competitive with respect to classical algorithms.

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arXiv2023

Splitting Vertices in 2-Layer Graph Drawings

Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines (layers), and their relationships (edges) are represented by segments connecting vertices. Methods for constructing 2-layer drawings often try to minimize the number of edge crossings. We use vertex splitting to reduce the number of crossings, by replacing selected vertices on one layer by two (or more) copies and suitably distributing their incident edges among these copies. We study several optimization problems related to vertex splitting, either minimizing the number of crossings or removing all crossings with fewest splits. While we prove that some variants are \NP-complete, we obtain polynomial-time algorithms for others. We run our algorithms on a benchmark set of bipartite graphs representing the relationships between human anatomical structures and cell types.

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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 $k$.

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