Proximity Drawings of High-Degree Trees

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set

On Vertex- and Empty-Ply Proximity Drawings

We initiate the study of the vertex-ply of straight-line drawings, as a relaxation of the recently introduced ply number. Consider the disks centered at each vertex with radius equal to half the length of the longest edge incident to the vertex. The vertex-ply of a drawing is determined by the vertex covered by the maximum number of disks. The main motivation for considering this relaxation is to relate the concept of ply to proximity drawings. In fact, if we interpret the set of disks as proximity regions, a drawing with vertex-ply number 1 can be seen as a weak proximity drawing, which we call empty-ply drawing. We show non-trivial relationships between the ply number and the vertex-ply number. Then, we focus on empty-ply drawings, proving some properties and studying what classes of graphs admit such drawings. Finally, we prove a lower bound on the ply and the vertex-ply of planar drawings.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

Mutual Witness Proximity Drawings of Isomorphic Trees

A pair $\langle G_0, G_1 \rangle$ of graphs admits a mutual witness proximity drawing $\langle \Gamma_0, \Gamma_1 \rangle$ when: (i) $\Gamma_i$ represents $G_i$, and (ii) there is an edge $(u,v)$ in $\Gamma_i$ if and only if there is no vertex $w$ in $\Gamma_{1-i}$ that is ``too close'' to both $u$ and $v$ ($i=0,1$). In this paper, we consider infinitely many definitions of closeness by adopting the $\beta$-proximity rule for any $\beta \in [1,\infty]$ and study pairs of isomorphic trees that admit a mutual witness $\beta$-proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness $\beta$-proximity drawing for any $\beta \in [1,\infty]$. The constructive technique can be made ``robust'': For some tree pairs we can suitably prune linearly many leaves from one of the two trees and still retain their mutual witness $\beta$-proximity drawability. Notably, in the special case of isomorphic caterpillars and $\beta=1$, we construct linearly separable mutual witness Gabriel drawings.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Reading the Transformations of an Urban Edge: From Liberty Era Palermo to the City of Today

To honour the battle of 27 May 1860, in 1910 the Palermo City Government decided to realise a commemorative monument. A position at the centre of a large circular plaza was of have afforded the monument a greater solemnity. The commission for the Monument was awarded to Ernesto Basile. In 1927 the City Government decided to dedicate the monument to the Fallen and asked Basile to complete the monument adding an architectural backdrop. The first version of the new project was a fence that enveloped the entire square and the ring road, interrupted only by entrances near the streets flowing into the square, and dividing it into four sectors. The final design instead called for the realisation of a semi-circular exedra of columns interrupted at the centre by a large gate that allows access to the square and to the back of the monument. The successive development of the city engulfed the square in the midst of tall and anonymous buildings realised, beginning in the 1960s, without any order of relations, stripping the surrounding fabric of its identity. Through the survey of the today‘s configuration, the analysis of Basile‘s original drawings and the representation of the modifications made over time, this text proposes an original reading of the configuration of Piazza Vittorio Veneto and the Monument to the Fallen, in relation to important moments in its history, from its design to the present day. The three-dimensional models reproduce the monument and its surroundings at the time of its construction in 1910, based on the first version for its expansion (unbuilt), with the addition of the exedra from 1930 and in its current condition. The redesign and extrapolation of different views of the digital models also provided original images of use to new readings of the perception of this space