Vincenzo Roselli

Morphing planar graph drawings with a polynomial number of steps

In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between the two drawings so that the drawing remains straight-line planar at all times. Cairnss original proof required exponentially many morphing steps. We prove that there is a morph that consists of O(n2) steps, where each step is a linear morph that moves each vertex at constant speed along a straight line. Using a known result on compatible triangulations this implies that for a general planar graph G and any two straight-line planar drawings of G with the same embedding, there is a morph between the two drawings that preserves straight-line planarity and consists of O(n4) steps.

Small point sets for simply-nested planar graphs

A point set P⊆ℝ2 is universal for a class

Morphing Planar Graph Drawings Optimally

\cal G

Relaxing the constraints of clustered planarity

if every graph of

The Importance of Being Proper

{\cal G}

Bitconeview: visualization of flows in the bitcoin transaction graph

has a planar straight-line embedding into P. We prove that there exists a

The importance of being proper

O(n (\frac{\log n}{\log\log n})^2)

Monotone drawings of graphs with fixed embedding

size universal point set for the class of simply-nested n-vertex planar graphs. This is a step towards a full answer for the well-known open problem on the size of the smallest universal point sets for planar graphs [1, 5, 9].

Drawing Planar Graphs with Many Collinear Vertices

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. Furthermore, 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.

Anchored Drawings of Planar Graphs

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, this 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 exists 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.