Outer 1-Planar Graphs

Abstract

A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in 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 N P-hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph G is o1p, then the algorithm computes an embedding and can augment G to a maximal o1p graph. Otherwise, G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) Grant Br835/18-1. B Christian Bachmaier

Gap-planar graphs

“…• Deciding whether a graph is 1-gap-planar is NP-complete, even when the drawing of a given graph is restricted to a fixed rotation system that is part of the input (Section 6). Note that analogous recognition problems for other families of beyond-planar graphs are also NP-hard (see, e.g., [7,10,14,15,31,43]), while polynomial algorithms are known in some restricted settings (see, e.g., [6,10,15,20,24,37,35]).…”

Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

“…Since outer-1-planar graphs are planar graphs [3], a planar orthogonal layout could be computed with curve complexity at most three. For example, in Fig.…”

“…These are the graphs that have a 1-planar drawing where every vertex lies on the outer (unbounded) face. They are planar graphs, can be recognized in linear time [3,12], and can be drawn with straight-line edges and right-angle crossings [8].…”

Recognizing Optimal 1-Planar Graphs in Linear Time

“…The resulting graphs are called outer 1-planar and were first studied by Eggleton [23]. It is not obvious that outer 1-planar graphs are planar [2]. Independently, Auer et al [2] and Hong et al [26] developed linear time recognition algorithms for outer 1-planar graphs.…”