4 research outputs found
Flips in combinatorial pointed pseudo-triangulations with face degree at most four
In this paper we consider the flip operation for combinatorial pointed pseudo-triangulations where faces have size 3 or 4, so-called combinatorial 4-PPTs. We show that every combinatorial 4-PPT is stretchable to a geometric pseudo-triangulation, which in general is not the case if faces may have size larger than 4. Moreover, we prove that the flip graph of combinatorial 4-PPTs with triangular outer face is connected and has diameter O(n2).European Science FoundationAustrian Science FundMinisterio de Ciencia e InnovaciĂłnJunta de Castilla y LeĂł
Flip Distance to a Non-crossing Perfect Matching
A perfect straight-line matching on a finite set of points in the
plane is a set of segments such that each point in is an endpoint of
exactly one segment. is non-crossing if no two segments in cross each
other. Given a perfect straight-line matching with at least one crossing,
we can remove this crossing by a flip operation. The flip operation removes two
crossing segments on a point set and adds two non-crossing segments to
attain a new perfect matching . It is well known that after a finite number
of flips, a non-crossing matching is attained and no further flip is possible.
However, prior to this work, no non-trivial upper bound on the number of flips
was known. If (resp.~) is the maximum length of the longest
(resp.~shortest) sequence of flips starting from any matching of size , we
show that and (resp.~ and
)
3-colorability of pseudo-triangulations
Electronic version of an article published as International Journal of Computational Geometry & Applications, Vol. 25, No. 4 (2015) 283–298 DOI: 10.1142/S0218195915500168 © 2015 World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijcgaDeciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.Postprint (author's final draft