Recognizing Outer 1-Planar Graphs in Linear Time

Abstract

Abstract.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 N P-hard.Our main result is a linear-time algorithm that first tests whether a graph 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.

Outer 1-Planar Graphs

“…The upper bound was also proved by Didimo [21] using a different approach. 5 2 n − 4 edges, and for every even n ≥ 2 there are o1p graphs with 5 2 n − 4 edges.…”

“…This is due to the fact that o1p graphs have an underlying tree structure, which finds expression in a simplified planar dual graph and results in treewidth at most three. The simplified dual of a maximal o1p graph is a ternary tree, whose nodes correspond to K 3 s and K 4 s. From these trees we obtain that every o1p graph of size n has at most 5 2 n − 4 edges and that there are sparse maximal o1p graphs with 11 5 n − 18 5 edges. The upper bound is n 2 − 1 above the respective value for outerplanar graphs.…”

Fan-planarity: Properties and complexity

“…• k-planar drawings, where an edge can have at most k crossings (see, e.g., [5,8,9,15,22,24,28,33,34,36,37,40]); • k-quasi-planar drawings, which do not contain k mutually crossing edges (see, e.g., [1,3,4,21,30,41]); • RAC (Right Angle Crossing) drawings, where edges can cross only at right angles (see, e.g., [25] and [26] for a survey); • ACE α drawings [2] and ACL α drawings [6,20,27], which are generalizations of RAC drawings; namely, in an ACE α drawing edges can cross only at an angle that is exactly α (α ∈ (0, π/2]); in an ACL α drawing edges can cross only at angles that are at least α (see also [26]);…”

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

“…This implies 1, 5b, and its symmetric counterpart. Moreover, all virtual edges have to be real edges which implies 2 and, combined with the previous observation, also 4. If the skeleton of an S-node would be a cycle of length greater than three, we could add chords, contradicting maximality.…”

“…Testing outer-1-planarity of a graph can be solved in linear time, as shown independently by Auer et al [4] and Hong et al [19]. It is worth to note that an outer-1-planar graph is always planar [4], while this is not true in general for outer-fan-planar graphs. Indeed, the complete graph K 5 is outer-fan-planar, but not planar.…”