Point--set embedding beyond
Point
planarity
Point--set embedding (PSE)
Point
• Definition. Let G be a planar graph and let S
be a set of points in the plane, a point-set
embedding of G on S is a planar drawing Γ of G
such that the vertices of G are represented in
Γ by the points of S
Point--set embedding (PSE)
Point
Point--set embedding (PSE)
Point
Point--set embedding (PSE)
Point
• Different variants depending on:
• the point set (arbitrary, general position, convex
position,...)
• the graph family (trees, outerplanar, planar,...)
• the type of drawing (straight-line, polyline,
orthogeodesic, upward,...)
• the mapping (no mapping, given mapping, partial
mapping)
Some results about PSE
• Outerplanar graphs are the largest family of
graphs that admit a straight-line PSE on every set
of points in general position
• Gritzmann, Mohar, Pach, and Pollack, AMM, 1991
• Bose, CGTA, 1997
• Planar graphs admit a PSE on every set of points
with at most 2 bends per edge
• Kaufmann & Wiese, JGAA, 2002
• Planar graphs admit a PSE with a given mapping
on every set of points with O(n) bends per edge
• Pach & Wenger, Graphs & Comb., 2001
k-planarity
• Definition. A graph is k-planar if it admits a
drawing where each edge is crossed at most k
times
1-planar
2-planar
Outer--k-planarity
Outer
• Definition. A outer-k-planar graph is a graph
that admits a k-planar drawing with all
vertices on the external face
Outer-1-planar graphs
can be recognized in
linear time
• Auer et al. , GD 2013
• Hong et al., GD 2013
Outer-1-planar
Two research directions
• Direction 1. Study k-planar PSE of planar
graphs
• overcome the limits of the planar case, at the
expenses of crossings
• Direction 2. Study k-planar PSE of k-planar
graphs
• find results analogous to the ones in the planar
case
Specific problems (direction 1)
• Problem 1.1. Do all planar graphs admit a kplanar PSE on every set of points with less
than 2 bends per edge?
• trade-offs bend/crossings?
• Problem 1.2. Do all planar graphs admit a kplanar PSE with a given mapping on every set
of points with o(n) bends per edge?
• trade-offs bend/crossings?
Specific problems (direction 2)
• Problem 1.1. Are outer-k-planar graphs the
largest family of graphs that admit a k-planar
straight-line PSE on every set of points in
general position?
• a good case could be k=1