Daniel Neuwirth

Recognizing Outer 1-Planar Graphs in Linear Time

A graph is outer 1-planar o1p if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1-planar graphs, whose recognition is

On a tree and a path with no geometric simultaneous embedding

\mathcal{NP}

NIC-planar graphs☆

-hard. Our main result is a linear-time algorithm that first tests whether a graphi¾?G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors see Fig. 3, which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p.

On Aligned Bar 1-Visibility Graphs

Two graphs G1 = (V,E1) and G2 = (V,E2) 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 G1 and for G2. The most prominent problem in this area is the question whether a tree and a path can always be simultaneously embedded. 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 prove that it is not always possible to simultaneously embed two edge-disjoint trees. Finally, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of height 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 height 4.

On Bar (1,j)-Visibility Graphs

A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per edge. We characterize embeddings of maximal NIC-planar graphs in terms of generalized planar dual graphs. The characterization is used to derive tight bounds on the density of maximal NIC-planar graphs which ranges between 3.2(n-2) and 3.6(n-2). Further, we prove that optimal NIC-planar graphs with 3.6(n-2) edges have a unique embedding and can be recognized in linear time, whereas the general recognition problem of NIC-planar graphs is NP-complete. In addition, we show that there are NIC-planar graphs that do not admit right angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.

On Some \mathcal{NP}-complete SEFE Problems

A graph is called a bar 1-visibility graph, if its vertices can be represented as horizontal vertex-segments, called bars, and each edge corresponds to a vertical line of sight which can traverse another bar. If all bars are aligned at one side, then the graph is an aligned bar 1-visibility graph, \(AB1V\) graph for short.

On a Tree and a Path with no Geometric Simultaneous Embedding

A graph is called a bar (1, j)-visibility graph if its vertices can be represented as horizontal vertex-segments (bars) and each edge as a vertical edge-segment connecting the bars of the end vertices such that each edge-segment intersects at most one other bar and each bar is intersected by at most j edge-segments. Bar (1, j)-visibility refines bar 1-visibility in which there is no bound on the number of intersections of bars.

Outer 1-Planar Graphs

We investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) problem which, given k planar graphs G 1,…,G k on the same set of vertices, asks whether they can be simultaneously embedded so that the embedding of each graph be planar and common edges be drawn the same. While the computational complexity of SEFE with k = 2 is a central open question in Graph Drawing, the problem is \(\mathcal{NP}\)-complete for k ≥ 3 [Gassner et al., WG ’06], even if the intersection graph is the same for each pair of graphs (sunflower intersection) [Schaefer, JGAA (2013)].