We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, with at most three bends per edge, where n is the number of vertices. The O(n) time algorithm guarantees that two corresponding vertices in the graphs are mapped to the same location in the final drawing and that both the drawings are crossing-free. The special case when both input graphs are trees has several applications, such as contour tree simplification and evolutionary biology. We show that if both the input graphs are are trees, only one bend per edge is required. The O(n) time algorithm guarantees that both drawings are crossings-free, corresponding tree vertices are mapped to the same locations, and all vertices (and bends) are on the O(n 2) × O(n 2) grid (O(n 3) × O(n 3) grid). For the special case when one of the graphs is a tree and the other is a path we can find simultaneous embedding with fixed-edges. That is, we can guarantee that corresponding vertices are mapped to the same locations and that corresponding edges are drawn the same way. We describe an O(n) time algorithm for simultaneous embedding with fixededges for tree-path pairs with at most one bend per tree-edge and no bends along path edges, such that all vertices (and bends) are on the O(n) × O(n 2) grid, (O(n 2) × O(n 3) grid).

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

Foundation (JU204/10-1). Abstract. We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a longstanding open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE. 1

Foundation (JU204/10-1). Abstract. We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that this problem is closely related to the weak realizability problem: Can a graph be drawn such that all edge crossings occur in a given set of edge pairs? By exploiting this relationship we can explain why the simultaneous embedding problem is challenging, both from a computational and a combinatorial point of view. More precisely, we prove that simultaneously embedding graphs with fixed edges is NP-complete even for three planar graphs. For two planar graphs the complexity status is still open. 1

A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective mapping of their vertex sets γ: V1 → V2 is a pair of planar straight-line drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2 = γ(v1) is mapped in Γ2 to the same point where v1 is mapped in Γ1, where v1 ∈ V1 and v2 ∈ V2. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that if the input graphs are assumed to share no common edges this does not seem to yield large classes of graphs that can be simultaneously embedded. Further, if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open for geometric simultaneous embedding. Finally, we present some positive and negative results on the near-simultaneous embedding problem, in which vertices are not forced to be placed exactly in the same, but just in “near” points in different drawings.

A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families—such as unlabeled level planar (ULP) graphs or the family of “carousel graphs”—are always matched drawable.

In this paper we consider the problem of characterizing the directed graphs that admit an upward straight-line embedding into every point set in convex or in general position. In particular, we show that no biconnected directed graph admits an upward straight-line embedding into every point set in convex position, and we provide a characterization of the Hamiltonian directed graphs that admit upward straight-line embeddings into every point set in general or in convex position. We also describe how to construct upward straight-line embeddings of directed paths into convex point sets and we prove that for directed trees such embeddings do not always exist. Further, we investigate the related problem of upward simultaneous embedding without mapping, proving that deciding whether two directed graphs admit an upward simultaneous embedding without mapping is NP-hard.

Graph augmentation problems are motivated by network design, and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems.

We consider characterizations of level planar trees. Healy et al. [8] characterized the set of trees that are level planar in terms of two minimal level non-planar (MLNP) patterns. Fowler and Kobourov [7] later proved that the set of patterns was incomplete and added two additional patterns. In this paper, we show that the characterization is still incomplete by providing new MLNP patterns not included in the previous characterizations. Moreover, we introduce an iterative method to create an arbitrary number of MLNP patterns, thus proving that the set of minimal patterns that characterizes level planar trees is infinite.

Two planar graphs G1 and G2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them, and the common subgraph is 2-connected.