Upward Planar Morphs

(left) A planar graph $G$ with a red and a blue cycle. (right) The skeleton of an R-node $\mu$ of the SPQR-tree of $G$ rooted at edge $e=uv$. The blue and red cycles are relevant for $\mu$ as they project to the blue and red cycle of the skeleton, respectively. The red cycle is an interface cycle

Abstract

We prove that, given two topologically equivalent upward planar straight-line drawings of an $n$-vertex directed graph $G$, there always exists a morph between them such that all the drawings of the morph are upward planar and straight-line. Such a morph consists of $O(1)$ morphing steps if $G$ is a reduced planar $st$-graph, $O(n)$ morphing steps if $G$ is a planar $st$-graph or a reduced upward planar graph, and $O(n^2)$ morphing steps if $G$ is a general upward planar graph.
Further, we show that there exist two topologically equivalent upward planar drawings such that any upward planar morph between them consists of $\Omega(n)$ linear morphing steps.

Publication
Algorithmica
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Giordano Da Lozzo
Assistant Professor (RTDa)

My research interests are in Algorithm Engineering and Complexity, focused in particular on the theoretical and algorithmic challenges arising from the visualization of graphs.

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