Planar Open Rectangle-of-Influence Drawings

A straight line drawing of a graph is an open weak rectangle-of-influence (RI) drawing, if there is no vertex in the relative interior of the axis parallel rectangle induced by the end points of each edge. Despite recent interest of the graph drawing community in rectangle-of-influence drawings, no algorithm is known to test whether a graph has a planar open weak RI-drawing, not even for inner triangulated graphs. In this thesis, we have two major contributions. First we study open weak RI-drawings of plane graphs that must have a non-aligned frame, i.e., the graph obtained from removing the interior of every filled triangle is drawn such that no two vertices have the same coordinate. We introduce a new way to assign labels to angles, i.e., instances of vertices on faces. Using this labeling, we provide necessary and sufficient conditions characterizing those plane graphs that have open weak RI-drawings with non-aligned frame. We also give a polynomial algorithm to construct such a drawing if one exists. Our second major result is a negative result: deciding if a planar graph (i.e., one where we can choose the planar embedding) has an open weak RI-drawing is NP-complete. NP-completeness holds even for open weak RI-drawings with non-aligned frames

Aligned Drawings of Planar Graphs

Let $G$ be a graph that is topologically embedded in the plane and let $\mathcal{A}$ be an arrangement of pseudolines intersecting the drawing of $G$. An aligned drawing of $G$ and $\mathcal{A}$ is a planar polyline drawing $\Gamma$ of $G$ with an arrangement $A$ of lines so that $\Gamma$ and $A$ are homeomorphic to $G$ and $\mathcal{A}$. We show that if $\mathcal{A}$ is stretchable and every edge $e$ either entirely lies on a pseudoline or it has at most one intersection with $\mathcal{A}$, then $G$ and $\mathcal{A}$ have a straight-line aligned drawing. In order to prove this result, we strengthen a result of Da Lozzo et al., and prove that a planar graph $G$ and a single pseudoline $\mathcal{L}$ have an aligned drawing with a prescribed convex drawing of the outer face. We also study the less restrictive version of the alignment problem with respect to one line, where only a set of vertices is given and we need to determine whether they can be collinear. We show that the problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

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

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

On Visibility Representations of Non-planar Graphs

A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge

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