arXiv subjects

Fabrizio Montecchiani

Publications and source records attributed to Fabrizio Montecchiani.

70 records · Page 4Linked to original sources

arXiv2017

Universal Slope Sets for 1-Bend Planar Drawings

We describe a set of $Δ-1$ slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree $Δ\geq 4$; this establishes a new upper bound of $Δ-1$ on the 1-bend planar slope number. By universal we mean that every planar graph of degree $Δ$ has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was $\frac{3}{2} (Δ-1)$ (the known lower bound being $\frac{3}{4} (Δ-1)$); secondly, all the known algorithms to construct 1-bend planar drawings with $O(Δ)$ slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is $\fracπ{(Δ-1)}$.

cs.CG↗

arXiv2016

On Partitioning the Edges of 1-Plane Graphs

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red-blue edge coloring of an optimal 1-plane graph $G$ partitions the edge set of $G$ into blue edges and red edges such that no two blue edges cross each other and no two red edges cross each other. We prove the following: $(i)$ Every optimal 1-plane graph has a red-blue edge coloring such that the blue subgraph is maximal planar while the red subgraph has vertex degree at most four; this bound on the vertex degree is worst-case optimal. $(ii)$ A red-blue edge coloring may not always induce a red forest of bounded vertex degree. Applications of these results to graph augmentation and graph drawing are also discussed.

cs.CG↗

arXiv2016

A Distributed Multilevel Force-directed Algorithm

The wide availability of powerful and inexpensive cloud computing services naturally motivates the study of distributed graph layout algorithms, able to scale to very large graphs. Nowadays, to process Big Data, companies are increasingly relying on PaaS infrastructures rather than buying and maintaining complex and expensive hardware. So far, only a few examples of basic force-directed algorithms that work in a distributed environment have been described. Instead, the design of a distributed multilevel force-directed algorithm is a much more challenging task, not yet addressed. We present the first multilevel force-directed algorithm based on a distributed vertex-centric paradigm, and its implementation on Giraph, a popular platform for distributed graph algorithms. Experiments show the effectiveness and the scalability of the approach. Using an inexpensive cloud computing service of Amazon, we draw graphs with ten million edges in about 60 minutes.

cs.DC↗

arXiv2016

Visibility Representations of Boxes in 2.5 Dimensions

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 only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $Γ$ is the set of bottom faces of the boxes in $Γ$.

cs.CG↗

arXiv2016

1-Bend RAC Drawings of 1-Planar Graphs

A graph is 1-planar if it has a drawing where each edge is crossed at most once. A drawing is RAC (Right Angle Crossing) if the edges cross only at right angles. The relationships between 1-planar graphs and RAC drawings have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC drawable and graphs that have a straight-line RAC drawing but that are not 1-planar. Also, straight-line RAC drawings always exist for IC-planar graphs, a subclass of 1-planar graphs. One of the main questions still open is whether every 1-planar graph has a RAC drawing with at most one bend per edge. We positively answer this question.

cs.CG↗

arXiv2016

1-bend Upward Planar Drawings of SP-digraphs

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. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawings with optimal angular resolution $\fracπΔ$. A variant of the proof technique is used to show that (non-directed) reduced series-parallel graphs and flat series-parallel graphs have a (non-upward) one-bend planar drawing with $\lceil\fracΔ{2}\rceil$ distinct slopes if biconnected, and with $\lceil\fracΔ{2}\rceil+1$ distinct slopes if connected.

cs.CG↗

arXiv2016

Placing Arrows in Directed Graph Drawings

We consider the problem of placing arrow heads in directed graph drawings without them overlapping other drawn objects. This gives drawings where edge directions can be deduced unambiguously. We show hardness of the problem, present exact and heuristic algorithms, and report on a practical study.

cs.DS↗

arXiv2016

Recognizing and Drawing IC-planar Graphs

IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph $G$ with $n$ vertices, we present an $O(n)$-time algorithm that computes a straight-line drawing of $G$ in quadratic area, and an $O(n^3)$-time algorithm that computes a straight-line drawing of $G$ with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar.

cs.CG↗

arXiv2016

A Distributed Force-Directed Algorithm on Giraph: Design and Experiments

In this paper we study the problem of designing a distributed graph visualization algorithm for large graphs. The algorithm must be simple to implement and the computing infrastructure must not require major hardware or software investments. We design, implement, and experiment a force-directed algorithm in Giraph, a popular open source framework for distributed computing, based on a vertex-centric design paradigm. The algorithm is tested both on real and artificial graphs with up to million edges, by using a rather inexpensive PaaS (Platform as a Service) infrastructure of Amazon. The experiments show the scalability and effectiveness of our technique when compared to a centralized implementation of the same force-directed model. We show that graphs with about one million edges can be drawn in less than 8 minutes, by spending about 1\$ per drawing in the cloud computing infrastructure.

cs.DS↗

arXiv2016

On Guarding Orthogonal Polygons with Sliding Cameras

A sliding camera inside an orthogonal polygon $P$ is a point guard that travels back and forth along an orthogonal line segment $γ$ in $P$. The sliding camera $g$ can see a point $p$ in $P$ if the perpendicular from $p$ onto $γ$ is inside $P$. In this paper, we give the first constant-factor approximation algorithm for the problem of guarding $P$ with the minimum number of sliding cameras. Next, we show that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth. Finally, we study art gallery theorems for sliding cameras, thus, give upper and lower bounds in terms of the number of guards needed relative to the number of vertices $n$.

cs.CG↗

arXiv2015

On Visibility Representations of Non-planar Graphs

A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

cs.CG↗

arXiv2015

L-Visibility Drawings of IC-planar Graphs

An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. We show that every IC-plane graph has a visibility drawing where every vertex is an L-shape, and every edge is either a horizontal or vertical segment. As a byproduct of our drawing technique, we prove that an IC-plane graph has a RAC drawing in quadratic area with at most two bends per edge.

cs.CG↗

arXiv2015

Simultaneous Visibility Representations of Plane st-graphs Using L-shapes

Let $\langle G_r,G_b \rangle$ be a pair of plane $st$-graphs with the same vertex set $V$. A simultaneous visibility representation with L-shapes of $\langle G_r,G_b \rangle$ is a pair of bar visibility representations $\langleΓ_r,Γ_b\rangle$ such that, for every vertex $v \in V$, $Γ_r(v)$ and $Γ_b(v)$ are a horizontal and a vertical segment, which share an end-point. In other words, every vertex is drawn as an $L$-shape, every edge of $G_r$ is a vertical visibility segment, and every edge of $G_b$ is a horizontal visibility segment. Also, no two L-shapes intersect each other. An L-shape has four possible rotations, and we assume that each vertex is given a rotation for its L-shape as part of the input. Our main results are: (i) a characterization of those pairs of plane $st$-graphs admitting such a representation, (ii) a cubic time algorithm to recognize them, and (iii) a linear time drawing algorithm if the test is positive.

cs.CG↗

arXiv2014

Algorithms and Bounds for Drawing Non-planar Graphs with Crossing-free Subgraphs

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.

cs.CG↗

arXiv2014

Properties and Complexity of Fan-Planarity

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 and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and $k$-planarity. Furthermore, we prove that deciding whether a graph admits a fan-planar drawing in the variable embedding setting is NP-complete.

cs.CG↗

arXiv2012

Progress on Partial Edge Drawings

Recently, a new way of avoiding crossings in straight-line drawings of non-planar graphs has been investigated. The idea of partial edge drawings (PED) is to drop the middle part of edges and rely on the remaining edge parts called stubs. We focus on a symmetric model (SPED) that requires the two stubs of an edge to be of equal length. In this way, the stub at the other endpoint of an edge assures the viewer of the edge's existence. We also consider an additional homogeneity constraint that forces the stub lengths to be a given fraction $δ$ of the edge lengths ($δ$-SHPED). Given length and direction of a stub, this model helps to infer the position of the opposite stub. We show that, for a fixed stub--edge length ratio $δ$, not all graphs have a $δ$-SHPED. Specifically, we show that $K_{241}$ does not have a 1/4-SHPED, while bandwidth-$k$ graphs always have a $Θ(1/\sqrt{k})$-SHPED. We also give bounds for complete bipartite graphs. Further, we consider the problem \textsc{MaxSPED} where the task is to compute the SPED of maximum total stub length that a given straight-line drawing contains. We present an efficient solution for 2-planar drawings and a 2-approximation algorithm for the dual problem.

cs.CG↗