216 research outputs found
Construction of planar 4-connected triangulations
In this article we describe a recursive structure for the class of 4-connected triangulations or - equivalently - cyclically 4-connected plane cubic graphs
Making triangulations 4-connected using flips
We show that any combinatorial triangulation on n vertices can be transformed
into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give
an example of an infinite family of triangulations that requires this many
flips to be made 4-connected, showing that our bound is tight. In addition, for
n >= 19, we improve the upper bound on the number of flips required to
transform any 4-connected triangulation into the canonical triangulation (the
triangulation with two dominant vertices), matching the known lower bound of 2n
- 15. Our results imply a new upper bound on the diameter of the flip graph of
5.2n - 33.6, improving on the previous best known bound of 6n - 30.Comment: 22 pages, 8 figures. Accepted to CGTA special issue for CCCG 2011.
Conference version available at
http://2011.cccg.ca/PDFschedule/papers/paper34.pd
Four-connected triangulations of planar point sets
In this paper, we consider the problem of determining in polynomial time
whether a given planar point set of points admits 4-connected
triangulation. We propose a necessary and sufficient condition for recognizing
, and present an algorithm of constructing a 4-connected
triangulation of . Thus, our algorithm solves a longstanding open problem in
computational geometry and geometric graph theory. We also provide a simple
method for constructing a noncomplex triangulation of which requires
steps. This method provides a new insight to the structure of
4-connected triangulation of point sets.Comment: 35 pages, 35 figures, 5 reference
Every 4-connected graph with crossing number 2 is Hamiltonian
A seminal theorem of Tutte states that 4-connected planar graphs are Hamiltonian. Applying a result of Thomas and Yu, one can show that every 4-connected graph with crossing number 1 is Hamiltonian. In this paper, we continue along this path and prove the titular statement. We also discuss the traceability and Hamiltonicity of 3-connected graphs with small crossing number and few 3-cuts, and present applications of our results
Circumference of essentially 4-connected planar triangulations
A -connected graph is essentially -connected if, for any -cut
of , at most one component of contains at least two
vertices. We prove that every essentially -connected maximal planar graph
on vertices contains a cycle of length at least ;
moreover, this bound is sharp
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