Algorithms for Incremental Planar Graph Drawing and Two-page Book Embeddings

Subject of this work are two problems related to ordering the vertices of planar graphs. The first one is concerned with the properties of vertex-orderings that serve as a basis for incremental drawing algorithms. Such a drawing algorithm usually extends a drawing by adding the vertices step-by-step as provided by the ordering. In the field of graph drawing several orderings are in use for this purpose. Some of them, however, lack certain properties that are desirable or required for classic incremental drawing methods. We narrow down these properties, and introduce the bitonic st-ordering, an ordering which combines the features only available when using canonical orderings with the flexibility of st-orderings. The additional property of being bitonic enables an st-ordering to be used in algorithms that usually require a canonical ordering. With this in mind, we describe a linear-time algorithm that computes such an ordering for every biconnected planar graph. Unlike canonical orderings, st-orderings extend to directed graphs, in particular planar st-graphs. Being able to compute bitonic st-orderings for planar st-graphs is of particular interest for upward planar drawing algorithms, since traditional incremental algorithms for undirected planar graphs might be adapted to directed graphs. Based on this observation, we give a full characterization of the class of planar st-graphs that admit such an ordering. This includes a linear-time algorithm for recognition and ordering. Furthermore, we show that by splitting specific edges of an instance that is not part of this class, one is able to transform it into one for which then such an ordering exists. To do so, we describe a linear-time algorithm for finding the smallest set of edges to split. We show that for a planar st-graph G=(V,E), |V|−3 edge splits are sufficient and every edge is split at most once. This immediately translates to the number of bends required for upward planar poly-line drawings. More specifically, we show that every planar st-graph admits an upward planar poly-line drawing in quadratic area with at most |V|−3 bends in total and at most one bend per edge. Moreover, the drawing can be obtained in linear time. The second part is concerned with embedding planar graphs with maximum degree three and four into books. Besides providing a simplified incremental linear-time algorithm for embedding triconnected 3-planar graphs into a book of two pages, we describe a linear-time algorithm to compute a subhamiltonian cycle in a triconnected 4-planar graph

Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

Let $G$ be an $n$-node planar graph. In a visibility representation of $G$, each node of $G$ is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of $G$ are vertically visible to each other. In the present paper we give the best known compact visibility representation of $G$. Given a canonical ordering of the triangulated $G$, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder's realizer for the triangulated $G$ yields a visibility representation of $G$ no wider than $\frac{22n-40}{15}$. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant's open question about whether $\frac{3n-6}{2}$ is a worst-case lower bound on the required width. Also, if $G$ has no degree-three (respectively, degree-five) internal node, then our visibility representation for $G$ is no wider than $\frac{4n-9}{3}$ (respectively, $\frac{4n-7}{3}$). Moreover, if $G$ is four-connected, then our visibility representation for $G$ is no wider than $n-1$, matching the best known result of Kant and He. As a by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Berlin, Germany, 200

Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.Comment: 37 page

Drawing Planar Graphs with Few Geometric Primitives

We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with $(8n-17)/3$ segments on an $O(n)\times O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with $3n/2$ edges on an $O(n)\times O(n^2)$ grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only $(5n - 11)/3$ arcs. This is significantly smaller than the lower bound of $2n$ for line segments for a nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017

Drawings of Planar Graphs with Few Slopes and Segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on $n$ vertices has a plane drawing with at most ${5/2}n$ segments and at most $2n$ slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.Comment: This paper is submitted to a journal. A preliminary version appeared as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. See http://arxiv.org/math/0606446 for a companion pape

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing, Bordeaux, 201

Compact Floor-Planning via Orderly Spanning Trees

Floor-planning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)-time algorithm to construct a floor-plan for any n-node plane triangulation. In comparison with previous floor-planning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces floor-plans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floor-plan area in a rectangle of size (n-1)x(2n+1)/3. Lower bounds on the worst-case area for floor-planning any plane triangulation are also provided in the paper.Comment: 13 pages, 5 figures, An early version of this work was presented at 9th International Symposium on Graph Drawing (GD 2001), Vienna, Austria, September 2001. Accepted to Journal of Algorithms, 200