Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices

A simultaneous embedding (with fixed edges) of two graphs $G^1$ and $G^2$ with common graph $G=G^1 \cap G^2$ is a pair of planar drawings of $G^1$ and $G^2$ that coincide on $G$. It is an open question whether there is a polynomial-time algorithm that decides whether two graphs admit a simultaneous embedding (problem SEFE). In this paper, we present two results. First, a set of three linear-time preprocessing algorithms that remove certain substructures from a given SEFE instance, producing a set of equivalent SEFE instances without such substructures. The structures we can remove are (1) cutvertices of the union graph $G^\cup = G^1 \cup G^2$, (2) most separating pairs of $G^\cup$, and (3) connected components of $G$ that are biconnected but not a cycle. Second, we give an $O(n^3)$-time algorithm solving SEFE for instances with the following restriction. Let $u$ be a pole of a P-node $μ$ in the SPQR-tree of a block of $G^1$ or $G^2$. Then at most three virtual edges of $μ$ may contain common edges incident to $u$. All algorithms extend to the sunflower case, i.e., to the case of more than three graphs pairwise intersecting in the same common graph.