Outerplanar graph drawings with few slopes

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that $\Delta-1$ edge slopes suffice for every outerplanar graph with maximum degree $\Delta\ge 4$. This improves on the previous bound of $O(\Delta^5)$, which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every $\Delta\ge 4$ there is an outerplanar graph with maximum degree $\Delta$ that requires at least $\Delta-1$ distinct edge slopes in an outerplanar straight-line drawing.Comment: Major revision of the whole pape

Planar Octilinear Drawings with One Bend Per Edge

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A $k$-planar graph is a planar graph in which each vertex has degree less or equal to $k$. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size $O(n^2) \times O(n)$. For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge

A Universal Slope Set for 1-Bend Planar Drawings

We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1)

Upward planar drawings with two slopes

In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes. We investigate whether a given upward planar digraph admits such a drawing and, if so, how to construct it. For the fixed embedding scenario, we give a simple characterisation and a linear-time construction by adopting algorithms from orthogonal drawings. For the variable embedding scenario, we describe a linear-time algorithm for single-source digraphs, a quartic-time algorithm for series-parallel digraphs, and a fixed-parameter tractable algorithm for general digraphs. For the latter two classes, we make use of SPQR-trees and the notion of upward spirality. As an application of this drawing style, we show how to draw an upward planar phylogenetic network with two slopes such that all leaves lie on a horizontal line

Drawing planar graphs with prescribed face areas

This thesis deals with planar drawings of planar graphs such that each interior face has a prescribed area. Our work is divided into two main sections. The rst one deals with straight-line drawings and the second one with orthogonal drawings. For straight-line drawings, it was known that such drawings exist for all planar graphs with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees, i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational coordinates if the face areas are rational, and we can bound the resolution. For orthogonal drawings, we give an algorithm to draw triconnected planar graphs with maximum degree 3. This algorithm produces a drawing with at most 8 bends per face and 4 bends per edge, which improves the previous known result of 34 bends per face. Both vertices and bends have rational coordinates if the face areas are rational

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi

Drawing Outer 1-planar Graphs with Few Slopes

A graph is outer 1-planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1-planar graphs are a superclass of the outerplanar graphs and a subclass of the partial 3-trees. We show that an outer 1-planar graph G of bounded degree Δ admits an outer 1-planar straight-line drawing that uses O(Δ) different slopes, which extends a previous result by Knauer et al. about the planar slope number of outerplanar graphs (CGTA, 2014). We also show that O(Δ2) slopes suffice to construct a crossing-free straight-line drawing of G; the best known upper bound on the planar slope number of planar partial 3-trees of bounded degree Δ is O(Δ5) and is proved by Jelínek et al. (Graphs and Combinatorics, 2013)