Giordano Da Lozzo

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. 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.

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, 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.

The Importance of Being Proper

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

On the relationship between k-planar and k-quasi planar graphs

\mathcal{NP}

The importance of being proper

-complete in the general case and that they become polynomial-time solvable when restricted to proper instances.

Intersection-Link Representations of Graphs

A graph is k-planar \((k \ge 1)\) if it can be drawn in the plane such that no edge is crossed \(k+1\) times or more. A graph is k-quasi-planar \((k \ge 2)\) if it can be drawn in the plane with no k pairwise crossing edges. The families of k-planar and k-quasi-planar graphs have been widely studied in the literature, and several bounds have been proven on their edge density. Nonetheless, only trivial results are known about the relationship between these two graph families. In this paper we prove that, for \(k \ge 3\), every k-planar graph is \((k+1)\)-quasi-planar.

Drawing Planar Graphs with Many Collinear Vertices

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.

Anchored Drawings of Planar 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 Ru and in which each edge u,i¾?v is represented by the intersection between Ru and Rv if it belongs to a dense subgraph or by a curve connecting the boundaries of Ru and Rv otherwise. We study a notion of planarity, called Clique Planarity, for intersection-link representations of graphs in which the dense subgraphs are cliques.

Clustered Planarity with Pipes

Given a planar graph G, what is the maximum number of collinear vertices in a planar straight-line drawing of G? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known: Every n-vertex planar graph has a planar straight-line drawing with \(\varOmega (\sqrt{n})\) collinear vertices; for every n, there is an n-vertex planar graph whose every planar straight-line drawinghas \(O(n^{0.986})\) collinear vertices; every n-vertex planar graph of treewidth at most two has a planar straight-line drawingwith \(\varTheta (n)\) collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs, partially answering two problems posed by Ravsky and Verbitsky. Similar results are not possible for all bounded treewidth or 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.

Beyond level planarity

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.