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

Abstract

Abstract. Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm to test whether a given graph is maximal outerfan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-hard, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given.

Recognizing Optimal 1-Planar Graphs in Linear Time

“…For 1-planarity this was proved by Grigoriev and Bodlaender [24] and by Korzhik and Mohar [28], and improved to hold for graphs of bounded bandwidth, pathwidth, or treewidth [4], for near planar graphs [15], and for 3-connected 1-planar graphs with a given rotation system [3]. Moreover, the recognition of right angle crossing graphs (RAC) [1] and of fan-planar graphs [5,6] is N P-hard. On the other hand, Eades et al [21] introduced a linear time testing algorithm for (planar) maximal 1-planar graphs that are given with a rotation system.…”

“…Proof If C R is good, then the degree vector of the four vertices that match the vertices of the inner cycle of CC is (4,4,5,5,5,5,6). The degree vector (5,5,5,5,5,6,6) implies a blocking red edge and −−→ H (x) = (4, 4, 5, 5, 6, 6, 6) implies a black edge between two opposite neighbors of a center, which violates CC.…”

“…None (3,4,4,5,5,5,6) Fixed (3,4,5,5,5,6,6) Partial (4,4,5,5,6,6,6) Fixed (5,5,5,5,5,5,6) Unclear Infeasible…”

“…A candidate x 1 of an optimal 1-planar graph is good for C R if and only if τ (x 1 ) = 4 and there are three more candidates 4,4,5,5,5,5,6), and CC matches the subgraph induced by x 1 , x 2 , x 3 , x 4 and its four common neighbors.…”

“…Independently, Auer et al [2] and Hong et al [26] developed linear time recognition algorithms for outer 1-planar graphs. Also, maximal outer-fan-planar graphs can be recognized in linear time [5]. Chen et al [17] developed a cubic-time recognition algorithm for hole-free 4-map graphs and observed that the 3-connected triangulated 1-planar graphs are exactly the 3-connected hole-free 4-map graphs [16].…”