7 research outputs found
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
Straightening out planar poly-line drawings
We show that any -monotone poly-line drawing can be straightened out while
maintaining -coordinates and height. The width may increase much, but we
also show that on some graphs exponential width is required if we do not want
to increase the height. Likewise -monotonicity is required: there are
poly-line drawings (not -monotone) that cannot be straightened out while
maintaining the height. We give some applications of our result.Comment: The main result turns out to be known (Pach & Toth, J. Graph Theory
2004, http://onlinelibrary.wiley.com/doi/10.1002/jgt.10168/pdf
B-VPG Representation of AT-free Outerplanar Graphs
B-VPG graphs are intersection graphs of axis-parallel line segments in
the plane. In this paper, we show that all AT-free outerplanar graphs are
B-VPG. We first prove that every AT-free outerplanar graph is an induced
subgraph of a biconnected outerpath (biconnected outerplanar graphs whose weak
dual is a path) and then we design a B-VPG drawing procedure for
biconnected outerpaths. Our proofs are constructive and give a polynomial time
B-VPG drawing algorithm for the class.
We also characterize all subgraphs of biconnected outerpaths and name this
graph class "linear outerplanar". This class is a proper superclass of AT-free
outerplanar graphs and a proper subclass of outerplanar graphs with pathwidth
at most 2. It turns out that every graph in this class can be realized both as
an induced subgraph and as a spanning subgraph of (different) biconnected
outerpaths.Comment: A preliminary version, which did not contain the characterization of
linear outerplanar graphs (Section 3), was presented in the
International Conference on Algorithms and Discrete Applied Mathematics
(CALDAM) 2022. The definition of linear outerplanar graphs in this paper
differs from that in the preliminary version and hence Section 4 is ne
2-Connecting outerplanar graphs without blowing up the pathwidth
Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl 1], in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two-dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl 1] for 2-vertex-connected outerplanar graphs will work for all outer planar graphs. (C) 2014 Elsevier B.V. All rights reserved