Extending Upward Planar Graph Drawings
The UPE-FUE problem is NP-complete. (Left) An instance $(G,\ell,\prec)$ of Partial Level Planarity with a prescribed upward embedding. (Center and right) An instance $(U,H,\Gamma_H)$ of UPE-FUE equivalent to $(G,\ell,\prec)$.
Abstract
In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes in input an upward planar drawing $\Gamma_H$ of a subgraph $H$ of a directed graph $G$ and asks whether $\Gamma_H$ can be extended to an upward planar drawing of $G$. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing.
We show the following results.
- First, we prove that the Upward Planarity Extension problem is NP-complete, even if $G$ has a prescribed upward embedding, the vertex set of $H$ coincides with the one of $G$, and $H$ contains no edge.
- Second, we show that the Upward Planarity Extension problem can be solved in $O(n \log n)$ time if $G$ is an $n$-vertex upward planar $st$-graph. This result improves upon a known $O(n^2)$-time algorithm, which however applies to all $n$-vertex single-source upward planar graphs.
- Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which $G$ is a directed path or cycle with a prescribed upward embedding, $H$ contains no edges, and no two vertices share the same $y$-coordinate in $\Gamma_H$.
Type
Publication
Computational Geometry: Theory and Applications
Giordano Da Lozzo
Assistant Professor (RTDb)
My research interests are in Algorithm Engineering and Complexity, focused in particular on the theoretical and algorithmic challenges arising from the visualization of graphs.