Transforming planar graph drawings while maintaining height

There are numerous styles of planar graph drawings, notably straight-line drawings, poly-line drawings, orthogonal graph drawings and visibility representations. In this note, we show that many of these drawings can be transformed from one style to another without changing the height of the drawing. We then give some applications of these transformations

Convex drawings of the complete graph: topology meets geometry

In this work, we introduce and develop a theory of convex drawings of the complete graph $K_n$ in the sphere. A drawing $D$ of $K_n$ is convex if, for every 3-cycle $T$ of $K_n$, there is a closed disc $\Delta_T$ bounded by $D[T]$ such that, for any two vertices $u,v$ with $D[u]$ and $D[v]$ both in $\Delta_T$, the entire edge $D[uv]$ is also contained in $\Delta_T$. As one application of this perspective, we consider drawings containing a non-convex $K_5$ that has restrictions on its extensions to drawings of $K_7$. For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal communication) showing that, for $n\le 12$, every optimal drawing of $K_n$ is convex. Convex drawings are characterized by excluding two of the five drawings of $K_5$. Two refinements of convex drawings are h-convex and f-convex drawings. The latter have been shown by Aichholzer et al (Deciding monotonicity of good drawings of the complete graph, Proc.~XVI Spanish Meeting on Computational Geometry (EGC 2015), 2015) and, independently, the authors of the current article (Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles, \rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}, to be equivalent to pseudolinear drawings. Also, h-convex drawings are equivalent to pseudospherical drawings as demonstrated recently by Arroyo et al (Extending drawings of complete graphs into arrangements of pseudocircles, submitted)

That Some of Sol Lewitt's Later Wall Drawings Aren't Wall Drawings

Sol LeWitt is probably most famous for wall drawings. They are an extension of work he had done in sculpture and on paper, in which a simple rule specifies permutations and variations of elements. With wall drawings, the rule is given for marks to be made on a wall. We should distinguish these algorithmic works from impossible-to-implement instruction works and works realized by following preparatory sketches. Taking the core feature of a wall drawing to be that it is algorithmic, some of LeWitt's later works are wall drawings in name only

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

That Some of Sol Lewitt's Later Wall Drawings Aren't Wall Drawings

Sol LeWitt is probably most famous for wall drawings. They are an extension of work he had done in sculpture and on paper, in which a simple rule specifies permutations and variations of elements. With wall drawings, the rule is given for marks to be made on a wall. We should distinguish these algorithmic works from impossible-to-implement instruction works and works realized by following preparatory sketches. Taking the core feature of a wall drawing to be that it is algorithmic, some of LeWitt's later works are wall drawings in name only

Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure