Fabrizio Frati

C-Planarity of C-Connected Clustered Graphs

We present the first characterization of c-planarity for c-connected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a linear-time c-planarity testing and embedding algorithm for c-connected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of well-known data structures as SPQR-trees and BC-trees. If the test fails, the algorithm identifies a structural element responsible for the non-cplanarity of the input clustered graph.

Efficient C-planarity testing for embedded flat clustered graphs with small faces

Let C be a clustered graph and suppose that the planar embedding of its underlying graph is fixed. Is testing the c-planarity of C easier than in the variable embedding setting? In this paper we give a first contribution towards answering the above question. Namely, we characterize c-planar embedded flat clustered graphs with at most five vertices per face and give an efficient testing algorithm for such graphs. The results are based on a more general methodology that shades new light on the c-planarity testing problem.

Small Area Drawings of Outerplanar Graphs

Abstract We show three linear-time algorithms for constructing planar straight-line grid drawings of outerplanar graphs. The first and the second algorithm are for balanced outerplanar graphs. Both require linear area. The drawings produced by the first algorithm are not outerplanar while those produced by the second algorithm are. On the other hand, the first algorithm constructs drawings with better angular resolution. The third algorithm constructs outerplanar drawings of general outerplanar graphs with O(n1.48) area. Further, we study the interplay between the area requirements of the drawings of an outerplanar graph and the area requirements of a special class of drawings of its dual tree.

On the perspectives opened by right angle crossing drawings

Right Angle Crossing (RAC) drawings are polyline drawings where each crossing forms four right angles. RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles. We investigate to what extent RAC drawings can help in overcoming the limitations of widely adopted planar graph drawing conventions, providing both positive and negative results. First, we prove that there exist acyclic planar digraphs not admitting any straight-line upward RAC drawing and that the corresponding decision problem is NP-hard. Also, we show digraphs whose straight-line upward RAC drawings require exponential area. Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions. We prove that every graph with vertex-degree at most 6 (at most 3) admits a RAC drawing with curve complexity 2 (resp. 1) and with quadratic area. Third, we consider a natural non-planar generalization of planar embedded graphs. Here we give bounds for curve complexity and area different from the ones known for planar embeddings.

On the Queue Number of Planar Graphs

We prove that planar graphs have

Embedding graphs simultaneously with fixed edges

O(\log^2 n)

Upward straight-line embeddings of directed graphs into point sets

queue number, thus improving upon the previous

A note on minimum-area straight-line drawings of planar graphs

O(\sqrt n)

On representing graphs by touching cuboids

upper bound. Consequently, planar graphs admit three-dimensional straight-line crossing-free grid drawings in

Succinct greedy drawings do not always exist

O(n \log^8 n)