Drawing complete multipartite graphs on the plane with restrictions on crossings

Abstract

We introduce the concept of NIC-planar graphs and present the full characterization of NIC-planar complete k-partite graphs.

NIC-planar graphs

“…Our N P-hardness result solves an open problem by Zhang [39] and can be obtained from known N P-hardness proofs [15,27], e. g., by a reduction from 1-planarity as in [15], which replaces every edge {u, v} of a graph G = (V, E) by the gadget in Fig. 16.…”

“…Lemma 10 suffices to prove upper and lower bounds on the density of maximal NIC-planar graphs. Whereas the upper bound of 18/5(n − 2) edges was already proven by Zhang [39] and shown to be tight by Czap andŠugerek [21], the lower bound has never been assessed before. Theorem 1.…”

Recognizing and drawing IC-planar graphs

“…An IC-planar graph is a graph that admits a 1-planar drawing where no two crossed edges share an endpoint, i.e., all crossing edges form a matching. ICplanar graphs have been mainly studied both in terms of their structure and in terms of their applications to coloring problems [4,40,53,54]. We prove that:…”

An annotated bibliography on 1-planarity

“…A 1-planar graph of class C 0 is also called an IC-planar graph, where IC stands for independent crossings (see, e.g., [5]). A 1-planar graph of class C 1 is also called a NIC-planar graph, where NIC stands for near-independent crossings (see, e.g., [135]). Czap anď Sugerek [49] observed that every 1-planar graph belongs to one of the classes C 0 , C 1 , or C 2 .…”