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

Diameter of the thick part of moduli space and simultaneous Whitehead moves

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the $\epsilon$-thick part of moduli space of S equipped with the Teichm\"uller or Thurston's Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order $\log \frac{g+p}{\epsilon}$. The same result also holds for the $\epsilon$-thick part of the moduli space of metric graphs of rank n equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrary labeled tree with n labels with simultaneous Whitehead moves, where the number of steps is of order log(n).Comment: 34 pages, 10 figures. Referee's comments incorporated. An appendix section is added to discuss the growth rate of the diameter of the space of graphs equipped with the metric of (non-simultaneous) Whitehead moves. The final version will appear in Duke Mathematical Journa

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

Reconfiguring Triangulations

The results in this thesis lie at the confluence of triangulations and reconfiguration. We make the observation that certain solved and unsolved problems about triangulations can be cast as reconfiguration problems. We then solve some reconfiguration problems that provide us new insights about triangulations. Following are the main contributions of this thesis: 1. We show that computing the flip distance between two triangulations of a point set is NP-complete. A flip is an operation that changes one triangulation into another by replacing one diagonal of a convex quadrilateral by the other diagonal. The flip distance, then, is the smallest number of flips needed to transform one triangulation into another. For the special case when the points are in convex position, the problem of computing the flip distance is a long-standing open problem. 2. Inspired by the problem of computing the flip distance, we start an investigation into computing shortest reconfiguration paths in reconfiguration graphs. We consider the reconfiguration graph of satisfying assignments of Boolean formulas where there is a node for each satisfying assignment of a formula and an edge whenever one assignment can be changed to another by changing the value of exactly one variable from 0 to 1 or from 1 to 0. We show that computing the shortest path between two satisfying assignments in the reconfiguration graph is either in P, NP-complete, or PSPACE-complete depending on the class the Boolean formula lies in. 3. We initiate the study of labelled reconfiguration. For the case of triangulations, we assign a unique label to each edge of the triangulation and a flip of an edge from e to e' assigns the same label to e' as e. We show that adding labels may make the reconfiguration graph disconnected. We also show that the worst-case reconfiguration distance changes when we assign labels. We show tight bounds on the worst case reconfiguration distance for edge-labelled triangulations of a convex polygon and of a spiral polygon, and edge-labelled spanning trees of a graph. We generalize the result on spanning trees to labelled bases of a matroid and show non-trivial upper bounds on the reconfiguration distance