28,728 research outputs found
The maximum size of adjacency-crossing graphs
An adjacency-crossing graph is a graph that can be drawn such that every two
edges that cross the same edge share a common endpoint. We show that the number
of edges in an -vertex adjacency-crossing graph is at most . If we
require the edges to be drawn as straight-line segments, then this upper bound
becomes . Both of these bounds are tight. The former result also follows
from a very recent and independent work of Cheong et al.\cite{cheong2023weakly}
who showed that the maximum size of weakly and strongly fan-planar graphs
coincide. By combining this result with the bound of Kaufmann and
Ueckerdt\cite{KU22} on the size of strongly fan-planar graphs and results of
Brandenburg\cite{Br20} by which the maximum size of adjacency-crossing graphs
equals the maximum size of fan-crossing graphs which in turn equals the maximum
size of weakly fan-planar graphs, one obtains the same bound on the size of
adjacency-crossing graphs. However, the proof presented here is different,
simpler and direct.Comment: 17 pages, 11 figure
On the size of planarly connected crossing graphs
We prove that if an -vertex graph can be drawn in the plane such that
each pair of crossing edges is independent and there is a crossing-free edge
that connects their endpoints, then has edges. Graphs that admit
such drawings are related to quasi-planar graphs and to maximal -planar and
fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
On the Number of Edges of Fan-Crossing Free Graphs
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1
if there are no k+1 edges , such that have a
common endpoint and crosses all . We prove a tight bound of 4n-8 on
the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9
bound for a straight-edge drawing. For k > 2, we prove an upper bound of
3(k-1)(n-2) edges. We also discuss generalizations to monotone graph
properties
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
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 outer-fan-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
- …