Edge partitions of optimal 2-plane and 3-plane graphs

Abstract

A topological graph is a graph drawn in the plane. A topological graph is k-plane, k > 0, if each edge is crossed at most k times. We study the problem of partitioning the edges of a k-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for k = 1, we focus on optimal 2-plane and 3-plane graphs, which are 2-plane and 3-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a forest, while (ii) an edge partition formed by a 1-plane graph and two plane forests always exists and can be computed in linear time. (iii) We describe efficient algorithms to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a plane graph with maximum vertex degree 12, or with maximum vertex degree 8 if the optimal 2-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) We exhibit an infinite family of simple optimal 2-plane graphs such that in any edge partition composed of a 1-plane graph and a plane graph, the plane graph has maximum vertex degree at least 6 and the 1-plane graph has maximum vertex degree at least 12.(v) We show that every optimal 3-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 2-plane graph and two plane forests.

Recognizing and Embedding Simple Optimal 2-Planar Graphs

“…The structure of simple optimal 2-planar graphs is well established. Namely, an optimal 2-planar rotation scheme R of a simple optimal 2-planar graph specifies a crossing-free 3-connected 3 spanning pentangulation P and a set of crossing edges [4,5]. The edges around each vertex are cyclically ordered so that every crossing-free edge (belonging to P) is followed by two crossing edges, which are succeeded by one crossing-free edge, and so on.…”

“…Assume w.l.o.g. that the edges of P 1 and P 1 incident to v 1 appear in this order and between edges (v 1 , w 1 ) and (v 1 , z) in the circular order of edges around v 1 4 ; refer to Fig. 11.…”

Efficiently Partitioning the Edges of a 1-Planar Graph into a Planar Graph and a Forest

“…Lenhart et al [18] show that optimal 1-planar graphs can be partitioned into a maximal planar graph and a planar graph of maximum degree four (the bound of four is shown to be optimal). Bekos et al [4] provide edge partition results for some k-planar graphs (graphs that can be drawn in the plane such that any edge crosses at most k other edges). Di Giacomo et al [8] prove edge partition results for so-called NIC-graphs, a subclass of 1-planar graphs.…”

Graphs Drawn With Some Vertices per Face: Density and Relationships

“…For example, it is known that n-vertex 1-planar graphs and 2-planar graphs have at most 4n − 8 edges and 5n − 10 edges, respectively, and both these bounds are tight, in the sense that there are graphs in these families that can actually achieve them [20], [52]. In the literature, a graph of F whose number of edges is the maximum possible over its number of vertices is usually called an optimal graph of F (see, e.g., [49], [53]- [57]).…”