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

B0_0-VPG Representation of AT-free Outerplanar Graphs

B$_0$-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$_0$-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$_0$-VPG drawing procedure for biconnected outerpaths. Our proofs are constructive and give a polynomial time B$_0$-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 $8^{th}$ 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

A Polynomial-time Algorithm for Outerplanar Diameter Improvement

The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.Comment: 24 page

Straightening out planar poly-line drawings

We show that any $y$-monotone poly-line drawing can be straightened out while maintaining $y$-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 $y$-monotonicity is required: there are poly-line drawings (not $y$-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

Upward and Orthogonal Planarity are W[1]-hard Parameterized by Treewidth

Upward planarity testing and Rectilinear planarity testing are central problems in graph drawing. It is known that they are both NP-complete, but XP when parameterized by treewidth. In this paper we show that these two problems are W[1]-hard parameterized by treewidth, which answers open problems posed in two earlier papers. The key step in our proof is an analysis of the All-or-Nothing Flow problem, a generalization of which was used as an intermediate step in the NP-completeness proof for both planarity testing problems. We prove that the flow problem is W[1]-hard parameterized by treewidth on planar graphs, and that the existing chain of reductions to the planarity testing problems can be adapted without blowing up the treewidth. Our reductions also show that the known $n^{O(tw)}$-time algorithms cannot be improved to run in time $n^{o(tw)}$ unless ETH fails.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023