Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal... more Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks. First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k > 2 and show that the problem is NP-complete for planar digraphs of maximum degree ∆ ≥ k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix...
We give an algorithm to compute a morph between any two convex drawings of the same plane graph. ... more 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.
ABSTRACT In this paper we study two problems related to the drawing of level graphs, that is, T-L... more ABSTRACT 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 -complete in the general case and that they become polynomial-time solvable when restricted to proper instances.
ABSTRACT In this paper we study the ANCHORED GRAPH DRAWING (AGD) problem: Given a planar graph G,... more ABSTRACT In this paper we study the ANCHORED GRAPH DRAWING (AGD) problem: Given a planar graph G, an initial placement for its vertices, and a distance d, produce a planar straight-line drawing of G such that each vertex is at distance at most d from its original position. We show that the AGD problem is NP-hard in several settings and provide a polynomial-time algorithm when d is the uniform distance L∞ and edges are required to be drawn as horizontal or vertical segments.
ABSTRACT In this work we investigate the complexity of some combinatorial problems related to the... more ABSTRACT In this work we investigate the complexity of some combinatorial problems related to the Simultaneous Embedding with Fixed Edges (SEFE) and the Partitioned T-Coherent k-Page Book Embedding (PTBE-k) problems, which are known to be equivalent under certain conditions. Given k planar graphs on the same set of n vertices, the SEFE problem asks to find a drawing of each graph on the same set of n points in such a way that each edge that is common to more than one graph is represented by the same curve in the drawings of all such graphs. Given a tree T with n leaves and a collection of k edge-sets connecting pairs of leaves of T, the PTBE-k problem asks to find an ordering of the leaves of T that is represented by T such that the endvertices of two edges in any set do not alternate in .
ABSTRACT We investigate the complexity of some problems related to the Simultaneous Embedding wit... more ABSTRACT We investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) problem which, given k planar graphs G 1,…,G k on the same set of vertices, asks whether they can be simultaneously embedded so that the embedding of each graph be planar and common edges be drawn the same. While the computational complexity of SEFE with k = 2 is a central open question in Graph Drawing, the problem is \(\mathcal{NP}\)-complete for k ≥ 3 [Gassner et al., WG ’06], even if the intersection graph is the same for each pair of graphs (sunflower intersection) [Schaefer, JGAA (2013)]. We improve on these results by proving that SEFE with k ≥ 3 and sunflower intersection is \(\mathcal{NP}\)-complete even when (i) the intersection graph is connected and (ii) two of the three input graphs are biconnected. This result implies that the Partitioned T-Coherent k-Page Book-Embedding is \(\mathcal{NP}\)-complete with k ≥ 3, which was only known for k unbounded [Hoske, Bachelor Thesis (2012)]. Further, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs is \(\mathcal{NP}\)-complete (optimization of SEFE, Open Problem 9, Chapter 11 of the Handbook of Graph Drawing and Visualization).
We present a system for the visualization of relational information on the smartphones. It is imp... more We present a system for the visualization of relational information on the smartphones. It is implemented on the iPhone and on the Google Android platforms and is based on a new visualization paradigm that poses interesting algorithmic challenges. We also show customizations of the system to explore and visualize popular social networks.
ABSTRACT Inter-domain routing data and Internet active probing measurements are two types of info... more ABSTRACT Inter-domain routing data and Internet active probing measurements are two types of information commonly available in huge datasets and subject to extensive, focused analysis. However, the study of the correlation between these two complementary types of information still remains one of the most challenging problems in today’s research in networking. In this paper we describe a metaphor for the visualization of the interplay between the routing information exchanged via BGP and the round-trip delay measurements collected by several geolocated probes. We implemented a prototype based on the above metaphor. Our prototype highlights both the Autonomous System topology and the latency associated with each AS-path over time. Further, it shows how probes are partitioned into clusters associated with each border gateway, based on observed traffic patterns. The resulting visualization allows the user to explore the dynamics of the correlation between the two types of information.
Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal... more Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks. First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k > 2 and show that the problem is NP-complete for planar digraphs of maximum degree ∆ ≥ k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix...
We give an algorithm to compute a morph between any two convex drawings of the same plane graph. ... more 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.
ABSTRACT In this paper we study two problems related to the drawing of level graphs, that is, T-L... more ABSTRACT 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 -complete in the general case and that they become polynomial-time solvable when restricted to proper instances.
ABSTRACT In this paper we study the ANCHORED GRAPH DRAWING (AGD) problem: Given a planar graph G,... more ABSTRACT In this paper we study the ANCHORED GRAPH DRAWING (AGD) problem: Given a planar graph G, an initial placement for its vertices, and a distance d, produce a planar straight-line drawing of G such that each vertex is at distance at most d from its original position. We show that the AGD problem is NP-hard in several settings and provide a polynomial-time algorithm when d is the uniform distance L∞ and edges are required to be drawn as horizontal or vertical segments.
ABSTRACT In this work we investigate the complexity of some combinatorial problems related to the... more ABSTRACT In this work we investigate the complexity of some combinatorial problems related to the Simultaneous Embedding with Fixed Edges (SEFE) and the Partitioned T-Coherent k-Page Book Embedding (PTBE-k) problems, which are known to be equivalent under certain conditions. Given k planar graphs on the same set of n vertices, the SEFE problem asks to find a drawing of each graph on the same set of n points in such a way that each edge that is common to more than one graph is represented by the same curve in the drawings of all such graphs. Given a tree T with n leaves and a collection of k edge-sets connecting pairs of leaves of T, the PTBE-k problem asks to find an ordering of the leaves of T that is represented by T such that the endvertices of two edges in any set do not alternate in .
ABSTRACT We investigate the complexity of some problems related to the Simultaneous Embedding wit... more ABSTRACT We investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) problem which, given k planar graphs G 1,…,G k on the same set of vertices, asks whether they can be simultaneously embedded so that the embedding of each graph be planar and common edges be drawn the same. While the computational complexity of SEFE with k = 2 is a central open question in Graph Drawing, the problem is \(\mathcal{NP}\)-complete for k ≥ 3 [Gassner et al., WG ’06], even if the intersection graph is the same for each pair of graphs (sunflower intersection) [Schaefer, JGAA (2013)]. We improve on these results by proving that SEFE with k ≥ 3 and sunflower intersection is \(\mathcal{NP}\)-complete even when (i) the intersection graph is connected and (ii) two of the three input graphs are biconnected. This result implies that the Partitioned T-Coherent k-Page Book-Embedding is \(\mathcal{NP}\)-complete with k ≥ 3, which was only known for k unbounded [Hoske, Bachelor Thesis (2012)]. Further, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs is \(\mathcal{NP}\)-complete (optimization of SEFE, Open Problem 9, Chapter 11 of the Handbook of Graph Drawing and Visualization).
We present a system for the visualization of relational information on the smartphones. It is imp... more We present a system for the visualization of relational information on the smartphones. It is implemented on the iPhone and on the Google Android platforms and is based on a new visualization paradigm that poses interesting algorithmic challenges. We also show customizations of the system to explore and visualize popular social networks.
ABSTRACT Inter-domain routing data and Internet active probing measurements are two types of info... more ABSTRACT Inter-domain routing data and Internet active probing measurements are two types of information commonly available in huge datasets and subject to extensive, focused analysis. However, the study of the correlation between these two complementary types of information still remains one of the most challenging problems in today’s research in networking. In this paper we describe a metaphor for the visualization of the interplay between the routing information exchanged via BGP and the round-trip delay measurements collected by several geolocated probes. We implemented a prototype based on the above metaphor. Our prototype highlights both the Autonomous System topology and the latency associated with each AS-path over time. Further, it shows how probes are partitioned into clusters associated with each border gateway, based on observed traffic patterns. The resulting visualization allows the user to explore the dynamics of the correlation between the two types of information.
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