16,649 research outputs found
Densities of Minor-Closed Graph Families
We define the limiting density of a minor-closed family of simple graphs F to
be the smallest number k such that every n-vertex graph in F has at most
kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting
densities. This set of numbers is countable, well-ordered, and closed; its
order type is at least {\omega}^{\omega}. It is the closure of the set of
densities of density-minimal graphs, graphs for which no minor has a greater
ratio of edges to vertices. By analyzing density-minimal graphs of low
densities, we find all limiting densities up to the first two cluster points of
the set of limiting densities, 1 and 3/2. For multigraphs, the only possible
limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure
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 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 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
Area density and regularity for soap film-like surfaces spanning graphs
For a given boundary set consisting of arcs and vertices, with two or more
arcs meeting at each vertex, we treat the problem of estimating the area
density of a soap film-like surface spanning the boundary.Comment: 32 page
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