Dynamic Embeddings of Dynamic Single-Source Upward Planar Graphs

A directed graph $G$ is upward planar if it admits a planar embedding such that each edge is $y$-monotone. Unlike planarity testing, upward planarity testing is NP-hard except in restricted cases, such as when the graph has the single-source property (i.e. each connected component only has one source). In this paper, we present a dynamic algorithm for maintaining a combinatorial embedding $\mathcal{E}(G)$ of a single-source upward planar graph subject to edge deletions, edge contractions, edge insertions upwards across a face, and single-source-preserving vertex splits through specified corners. We furthermore support changes to the embedding $\mathcal{E}(G)$ on the form of subgraph flips that mirror or slide the placement of a subgraph that is connected to the rest of the graph via at most two vertices. All update operations are supported as long as the graph remains upward planar, and all queries are supported as long as the graph remains single-source. Updates that violate upward planarity are identified as such and rejected by our update algorithm. We dynamically maintain a linear-size data structure on $G$ which supports incidence queries between a vertex and a face, and upward-linkability of vertex pairs. If a pair of vertices are not upwards-linkable, we facilitate one-flip-linkable queries that point to a subgraph flip that makes them linkable, if any such flip exists. We support all updates and queries in $O(\log^2 n)$ time.