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
