On a Tree and a Path with no Geometric Simultaneous Embedding

Two graphs $G_1=(V,E_1)$ and $G_2=(V,E_2)$ admit a geometric simultaneous embedding if there exists a set of points P and a bijection M: P -> V that induce planar straight-line embeddings both for $G_1$ and for $G_2$. While it is known that two caterpillars always admit a geometric simultaneous embedding and that two trees not always admit one, the question about a tree and a path is still open and is often regarded as the most prominent open problem in this area. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. As a final result, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of depth 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has depth 4.Comment: 42 pages, 33 figure

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi: V \to P$ such that the induced straight-line drawings of $G_1,\dots,G_k$ under $\phi$ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for $\exists\mathbb{R}$, the existential theory of the reals. Hence the problem $k$-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to $k$-SGE, with the property that both numbers $k$ and $n$ are linear in the number of pseudolines. This implies not only the $\exists\mathbb{R}$-hardness result, but also a $2^{2^{\Omega (n)}}$ lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only $2^{2^{\Omega (\sqrt{n})}}$

Multi-Perspective, Simultaneous Embedding

We describe MPSE: a Multi-Perspective Simultaneous Embedding method for visualizing high-dimensional data, based on multiple pairwise distances between the data points. Specifically, MPSE computes positions for the points in 3D and provides different views into the data by means of 2D projections (planes) that preserve each of the given distance matrices. We consider two versions of the problem: fixed projections and variable projections. MPSE with fixed projections takes as input a set of pairwise distance matrices defined on the data points, along with the same number of projections and embeds the points in 3D so that the pairwise distances are preserved in the given projections. MPSE with variable projections takes as input a set of pairwise distance matrices and embeds the points in 3D while also computing the appropriate projections that preserve the pairwise distances. The proposed approach can be useful in multiple scenarios: from creating simultaneous embedding of multiple graphs on the same set of vertices, to reconstructing a 3D object from multiple 2D snapshots, to analyzing data from multiple points of view. We provide a functional prototype of MPSE that is based on an adaptive and stochastic generalization of multi-dimensional scaling to multiple distances and multiple variable projections. We provide an extensive quantitative evaluation with datasets of different sizes and using different number of projections, as well as several examples that illustrate the quality of the resulting solutions