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

Leftist Canonical Ordering

Canonical ordering is an important tool in planar graph drawing and other applications. Although a linear-time algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering

More Canonical Ordering

Canonical ordering is an important tool in planar graph drawing and other applications. Although a linear-time algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering. Further, we discuss duality aspects and relations to Schnyder woods. Submitted

Kombinatorische Konzepte und Algorithmen zum Zeichnen planarer Graphen

In this thesis, we consider properties of triconnected, planar graphs and devote ourselves to the concepts canonical ordering and Schnyder woods. Whereas these concepts mostly have been investigated separately so far, we focus this thesis on their similarities and connection points, and present them consistently for the first time.The title "Combinatorial Concepts and Algorithms for Drawing Planar Graphs" already reflects the twofold structure of this thesis.In Part I "Concepts" we introduce, after giving a summary of fundamental properties of planar graphs, the combinatorial concepts of canonical ordering and Schnyder woods. Furthermore, we reveal their connection to triangle contact representations.Canonical ordering and Schnyder woods have manifold applications in graph encoding, in dimension theory, in the area of counting various kinds of planar maps, and more.We concentrate on graph drawing, in particular on triangle contact representations, and present in Part II "Algorithms" appropriate detailed and consistent algorithms with focus on the implementation of the leftist canonical ordering.Main contributions of this thesis are the introduction of the concept of leftist canonical ordering, the establishing of a connection via ordered path partitions to minimal Schnyder woods, and the provision of efficient algorithms with detailed pseudocodes that ease coding.Based on these, we show how and on which graph classes these methods can be used to determine homothetic triangle contact representations. Furthermore, we give an extensive overview of related concepts in a consistent manner and provide various new, simpler or more consistent proofs and algorithms