Upward planarity testing of embedded mixed graphs
GD'11 Proceedings of the 19th international conference on Graph Drawing
Upward planarity testing via SAT
GD'12 Proceedings of the 20th international conference on Graph Drawing
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A digraph is upward planar if it admits a planar drawing where all edges are monotone in the upward direction. It is known that the problem of testing a digraph for upward planarity is NP-complete in general. This paper describes an $O(n^4)$-time upward planarity testing algorithm for all digraphs that have a series-parallel structure, where $n$ is the number of vertices of the input. This significantly enlarges the family of digraphs for which a polynomial-time testing algorithm is known. Furthermore, the study is extended to general digraphs, and a fixed parameter tractable algorithm for upward planarity testing is described, whose time complexity is $O(d^t \cdot t \cdot n^3 + d \cdot t^2 \cdot n + d^2 \cdot n^2)$ where $t$ is the number of triconnected components of the digraph and $d$ is an upper bound on the diameter of any split component of the digraph. Our results use the new notion of upward spirality that, informally speaking, is a measure of the “level of winding” that a triconnected component of a digraph $G$ can have in an upward planar drawing of $G$.