Reconfiguring Triangulations

Abstract

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.