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Flip graphs of combinatorial and geometric objects are at the heart of manydeep structural insights and connections between different branches of discretemathematics and computer science. They also provide a natural framework for thestudy of reconfiguration problems. We study flip graphs of arrangements ofpseudolines and of arrangements of pseudocircles, which are combinatorialgeneralizations of lines and circles, respectively. In both cases we considertriangle flips as local transformation and prove conjectures regarding theirconnectivity.  In the case of $n$ pseudolines we show that the connectivity of the flipgraph equals its minimum degree, which is exactly $n-2$. For the proof weintroduce the class of shellable line arrangements, which serve as referenceobjects for the construction of disjoint paths. In fact, shellable arrangementsare elements of a flip graph of line arrangements which are vertices of apolytope (Felsner and Ziegler; DM 241 (2001), 301--312). This polytope forms acluster of good connectivity in the flip graph of pseudolines. In the case ofpseudocircles we show that triangle flips induce a connected flip graph on\emph{intersecting} arrangements and also on cylindrical intersectingarrangements. The result for cylindrical arrangements is used in the proof forintersecting arrangements. We also show that in both settings the diameter ofthe flip graph is in $\Theta(n^3)$. Our constructions make essential use ofvariants of the sweeping lemma for pseudocircle arrangements (Snoeyink andHershberger; Proc.\ SoCG 1989: 354--363). We finally study cylindricalarrangements in their own right and provide new combinatorial characterizationsof this class.

Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles