Simultaneous Embeddings with Few Bends and Crossings

A simultaneous embedding with fixed edges (SEFE) of two planar graphs $R$ and $B$ is a pair of plane drawings of $R$ and $B$ that coincide when restricted to the common vertices and edges of $R$ and $B$. We show that whenever $R$ and $B$ admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if $R$ and $B$ are trees then one bend per edge and four crossings per edge pair suffice (and one bend per edge is sometimes necessary), (2) if $R$ is a planar graph and $B$ is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if $R$ and $B$ are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. Our results improve on a paper by Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge pair suffice.Comment: Full version of the paper "Simultaneous Embeddings with Few Bends and Crossings" accepted at GD '1

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

Planar projections of graphs

We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices has a representation in $\lceil \sqrt{n/2}+1 \rceil$ planes. In 3 dimensions, we show that there exist graphs with $6n-15$ edges that can be projected onto two orthogonal planes, and that this is best possible. Finally, we obtain bounds in terms of parameters such as geometric thickness and linear arboricity. Using such a bound, we show that every graph of maximum degree 5 has a plane-projectable representation in 3 dimensions.Comment: Accepted at CALDAM 202

Simultaneous Orthogonal Planarity

We introduce and study the $\textit{OrthoSEFE}-k$ problem: Given $k$ planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the $k$ graphs? We show that the problem is NP-complete for $k \geq 3$ even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for $k \geq 2$ even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for $k=2$ when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Characterization of Unlabeled Radial Level Planar Graphs

Suppose that an n-vertex graph has a distinct labeling with the integers {1, . . . , n}. Such a graph is radial level planar if it admits a crossings-free drawing under two constraints. First, each vertex lies on a concentric circle such that the radius of the circle equals the label of the vertex. Second, each edge is drawn with a radially monotone curve. We characterize the set of unlabeled radial level planar (URLP) graphs that are radial level planar in terms of 7 and 15 forbidden subdivisions depending on whether the graph is disconnected or connected, respectively. We also provide linear-time drawing algorithms for any URLP graph

Column Planarity and Partial Simultaneous Geometric Embedding

We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any ε > 0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6 + ε)n. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partial SGE (PSGE). A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular, we show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE

Graph Simultaneous Embedding Tool, GraphSET

Problems in simultaneous graph drawing involve the layout of several graphs on a shared vertex set. This paper describes a Graph Simultaneous Embedding Tool, GraphSET, designed to allow the investigation of a wide range of embedding problems. GraphSET can be used in the study of several variants of simultaneous embedding including simultaneous geometric embedding, simultaneous embedding with fixed edges and colored simultaneous embedding with the vertex set partitioned into color classes. The tool has two primary uses: (i) studying theoretical problems in simultaneous graph drawing through the production of examples and counterexamples and (ii) producing layouts of given classes of graphs using built-in implementations of known algorithms. GraphSET along with movies illustrating its utility are available a

Geometric Simultaneous Embeddings of a Graph and a Matching

The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exists a planar graph and a matching which do not admit a geometric simultaneous embedding. This generalizes the same result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our proof for a matching and a tree sheds new light on a major open question: do a tree and a path always admit a geometric simultaneous embedding? Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations