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Fabrizio Frati

Publications and source records attributed to Fabrizio Frati.

At least 55 records · Page 3Linked to original sources
arXiv2016

Computing NodeTrix Representations of Clustered Graphs

NodeTrix representations are a popular way to visualize clustered graphs; they represent clusters as adjacency matrices and inter-cluster edges as curves connecting the matrix boundaries. We study the complexity of constructing NodeTrix representations focusing on planarity testing problems, and we show several NP-completeness results and some polynomial-time algorithms. Building on such algorithms we develop a JavaScript library for NodeTrix representations aimed at reducing the crossings between edges incident to the same matrix.

cs.CG↗
arXiv2016

Drawing Planar Graphs with Many Collinear Vertices

Consider the following problem: Given a planar graph $G$, what is the maximum number $p$ such that $G$ has a planar straight-line drawing with $p$ collinear vertices? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known for it: Every $n$-vertex planar graph has a planar straight-line drawing with $Ω(\sqrt{n})$ collinear vertices; for every $n$, there is an $n$-vertex planar graph whose every planar straight-line drawing has $O(n^σ)$ collinear vertices, where $σ<0.986$; every $n$-vertex planar graph of treewidth at most two has a planar straight-line drawing with $Θ(n)$ collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs. This (partially) answers two open problems posed by Ravsky and Verbitsky [WG 2011:295--306]. Similar results are not possible for all bounded treewidth planar graphs or for all bounded degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.

cs.CG↗
arXiv2016

Beyond Level Planarity

In this paper we settle the computational complexity of two open problems related to the extension of the notion of level planarity to surfaces different from the plane. Namely, we show that the problems of testing the existence of a level embedding of a level graph on the surface of the rolling cylinder or on the surface of the torus, respectively known by the name of $\textit{Cyclic Level Planarity}$ and $\textit{Torus Level Planarity}$, are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the $\textit{Simultaneous Level Planarity}$ of a set of level graphs, with respect to both the number of level graphs and the number of levels.

cs.DS↗
arXiv2016

Stack and Queue Layouts via Layered Separators

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed $g$ and $k$, we show that every $n$-vertex graph that can be embedded on a surface of genus $g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$; this includes $k$-planar graphs. The previously best known bound for the stack-number of these families was $\mathcal{O}(\sqrt{n})$, except in the case of $1$-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that $n$-vertex graphs that admit constant layered separators have $\mathcal{O}(\log n)$ stack-number.

cs.CG↗
arXiv2016

How to morph planar graph drawings

Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of $O(n)$ steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns' 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps.

cs.CG↗
arXiv2015

Simultaneous Embeddings with Few Bends and Crossings

A simultaneous embedding with fixed edges (SEFE) of two planar graphs $R$ and $B$ is a pair of plane drawings of $R$ and $B$ that coincide when restricted to the common vertices and edges of $R$ and $B$. We show that whenever $R$ and $B$ admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if $R$ and $B$ are trees then one bend per edge and four crossings per edge pair suffice (and one bend per edge is sometimes necessary), (2) if $R$ is a planar graph and $B$ is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if $R$ and $B$ are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. Our results improve on a paper by Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge pair suffice.

cs.CG↗
arXiv2015

Intersection-Link Representations of Graphs

We consider drawings of graphs that contain dense subgraphs. We introduce intersection-link representations for such graphs, in which each vertex $u$ is represented by a geometric object $R(u)$ and in which each edge $(u,v)$ is represented by the intersection between $R(u)$ and $R(v)$ if it belongs to a dense subgraph or by a curve connecting the boundaries of $R(u)$ and $R(v)$ otherwise. We study a notion of planarity, called Clique Planarity, for intersection-link representations of graphs in which the dense subgraphs are cliques.

cs.DS↗
arXiv2015

A Lower Bound on the Diameter of the Flip Graph

The flip graph is the graph whose nodes correspond to non-isomorphic combinatorial triangulations and whose edges connect pairs of triangulations that can be obtained one from the other by flipping a single edge. In this note we show that the diameter of the flip graph is at least $\frac{7n}{3} + Θ(1)$, improving upon the previous $2n + Θ(1)$ lower bound.

cs.CG↗
arXiv2015

Optimal Morphs of Convex Drawings

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.

cs.CG↗
arXiv2014

Drawing Partially Embedded and Simultaneously Planar Graphs

We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and the simultaneous planarity problem---to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed.

cs.CG↗
arXiv2014

Increasing-Chord Graphs On Point Sets

We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P. Further, we prove that, for every convex point set P with n points, there exists an increasing-chord graph with O(n log n) edges (and with no Steiner points) spanning P.

cs.CG↗
arXiv2014

Compatible Connectivity-Augmentation of Planar Disconnected Graphs

Motivated by applications to graph morphing, we consider the following \emph{compatible connectivity-augmentation problem}: We are given a labelled $n$-vertex planar graph, $\mathcal{G}$, that has $r\ge 2$ connected components, and $k\ge 2$ isomorphic planar straight-line drawings, $G_1,\ldots,G_k$, of $\mathcal{G}$. We wish to augment $\mathcal G$ by adding vertices and edges to make it connected in such a way that these vertices and edges can be added to $G_1,\ldots,G_k$ as points and straight-line segments, respectively, to obtain $k$ planar straight-line drawings isomorphic to the augmentation of $\mathcal G$. We show that adding $Θ(nr^{1-1/k})$ edges and vertices to $\mathcal{G}$ is always sufficient and sometimes necessary to achieve this goal. The upper bound holds for all $r\in\{2,\ldots,n\}$ and $k\ge 2$ and is achievable by an algorithm whose running time is $O(nr^{1-1/k})$ for $k=O(1)$ and whose running time is $O(kn^2)$ for general values of $k$. The lower bound holds for all $r\in\{2,\ldots,n/4\}$ and $k\ge 2$.

cs.CG↗
arXiv2014

On the Complexity of Clustered-Level Planarity and T-Level Planarity

In this paper we study two problems related to the drawing of level graphs, that is, T-LEVEL PLANARITY and CLUSTERED-LEVEL PLANARITY. We show that both problems are NP-complete in the general case and that they become polynomial-time solvable when restricted to proper instances.

cs.DS↗
arXiv2014

Morphing Planar Graph Drawings Optimally

We provide an algorithm for computing a planar morph between any two planar straight-line drawings of any $n$-vertex plane graph in $O(n)$ morphing steps, thus improving upon the previously best known $O(n^2)$ upper bound. Further, we prove that our algorithm is optimal, that is, we show that there exist two planar straight-line drawings $Γ_s$ and $Γ_t$ of an $n$-vertex plane graph $G$ such that any planar morph between $Γ_s$ and $Γ_t$ requires $Ω(n)$ morphing steps.

cs.DS↗
arXiv2014

Relaxing the Constraints of Clustered Planarity

In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, such a condition is not sufficient and the consequences on the problem due to the requirement of not having edge-region and region-region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge-edge, edge-region, or region-region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge-edge, edge-region, and region-region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge-edge or with only edge-region crossings always exist, while drawings with only region-region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region-region crossings exist for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs.

cs.DM↗
arXiv2013

Augmenting graphs to minimize the diameter

We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.

cs.DS↗