Giuseppe Liotta

Ortho-Polygon Visibility Representations of Embedded Graphs

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. Namely, we prove that if G is 1-plane (i.e., it has at most one crossing per edge) an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute in O(n) time an OPVR of G whose vertex complexity is bounded by a constant. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be (varOmega (n)). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed. Finally, we present the results of an experimental study on the vertex complexity of OPVRs of 1-plane graphs.

Ortho-polygon Visibility Representations of Embedded Graphs

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding-preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. More precisely, we prove that if G has at most one crossing per edge (i.e., G is a 1-plane graph), an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute an OPVR of G whose vertex complexity is bounded by a constant in O(n) time. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be

A Distributed Multilevel Force-Directed Algorithm

varOmega (n)

Visibility representations of boxes in 2.5 dimensions

Ω(n). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed in O(n) time. Finally, we present the results of an experimental study on the vertex complexity of ortho-polygon visibility representations of 1-plane graphs.

The Partial Visibility Representation Extension Problem

The ply number of a drawing is a new criterion of interest for graph drawing. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the longest edge incident to that vertex. This paper reports the results of an extensive experimental study that attempts to estimate correlations between the ply numbers and other aesthetic quality metrics for a graph layout, such as stress, edge-length uniformity, and edge crossings. We also investigate the performances of several graph drawing algorithms in terms of ply number, and provides new insights on the theoretical gap between lower and upper bounds on the ply number of k-ary trees.

Visibility Representations of Boxes in 2.5 Dimensions

The use of graph visualization approaches to present and analyze complex data is taking a leading role in conveying information and knowledge to users in many application domains. This creates the need of developing efficient and effective algorithms that automatically compute graph layouts. In this respect, force-directed algorithms are arguably among the most popular graph layout techniques. Aimed at leveraging the potential of modern distributed graph algorithms platforms, we present Multi-GiLA, the first multilevel force-directed graph visualization algorithm based on a vertex-centric computation paradigm. We implemented Multi-GiLA using the Apache Giraph platform. Experiments show that it can be successfully applied to compute high quality layouts of very large graphs on inexpensive cloud computing platforms.

1-Bend Upward Planar Drawings of SP-Digraphs

A graph is 1-planar if it has a drawing where each edge is crossed at most once. A drawing is RAC (Right Angle Crossing) if the edges cross only at right angles. The relationships between 1-planar graphs and RAC drawings have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC drawable and graphs that have a straight-line RAC drawing but that are not 1-planar [22]. Also, straight-line RAC drawings always exist for IC-planar graphs [9], a subclass of 1-planar graphs. One of the main questions still open is whether every 1-planar graph has a RAC drawing with at most one bend per edge. We positively answer this question.

Monotone Simultaneous Embeddings of Paths in d Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane (z=0) and edges are unobstructed lines of sight parallel to the x- or y-axis. We prove that: (i) Every complete bipartite graph admits a 2.5D-BR; (ii) The complete graph (K_n) admits a 2.5D-BR if and only if (n leqslant 19); (iii) Every graph with pathwidth at most 7 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an n-vertex graph that admits a 2.5D-GBR has at most (4n - 6 sqrt{n}) edges and this bound is tight. Finally, we prove that deciding whether a given graph G admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR (varGamma ) is the set of bottom faces of the boxes in (varGamma ).