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A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per edge. We characterize embeddings of maximal NIC-planar graphs in terms of generalized planar dual graphs. The characterization is used to derive tight bounds on the density of maximal NIC-planar graphs which ranges between 3.2(n-2) and 3.6(n-2). Further, we prove that optimal NIC-planar graphs with 3.6(n-2) edges have a unique embedding and can be recognized in linear time, whereas the general recognition problem of NIC-planar graphs is NP-complete. In addition, we show that there are NIC-planar graphs that do not admit right angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.
We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et. al. [3]. Given an undirected graph with a source $s$ and a destination $t$, find the smallest subset of vertices whose intersection with any $s-t$ path re
A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing edges to sh
Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measur
A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and vs neighbors. Such colorings have applications in wirel
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