7 research outputs found
A Note on Flips in Diagonal Rectangulations
Rectangulations are partitions of a square into axis-aligned rectangles. A
number of results provide bijections between combinatorial equivalence classes
of rectangulations and families of pattern-avoiding permutations. Other results
deal with local changes involving a single edge of a rectangulation, referred
to as flips, edge rotations, or edge pivoting. Such operations induce a graph
on equivalence classes of rectangulations, related to so-called flip graphs on
triangulations and other families of geometric partitions. In this note, we
consider a family of flip operations on the equivalence classes of diagonal
rectangulations, and their interpretation as transpositions in the associated
Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This
complements results from Law and Reading (JCTA, 2012) and provides a complete
characterization of flip operations on diagonal rectangulations, in both
geometric and combinatorial terms
Some notes on generic rectangulations
A rectangulation is a subdivision of a rectangle into rectangles. A generic rectangulation is a rectangulation that has no crossing segments. We explain several observations and pose some questions about generic rectangulations. In particular, we show how one may "centrally invert" a generic rectangulation about any given rectangle, analogous to reflection across a circle in classical geometry. We also explore 3-dimensional orthogonal polytopes related to "marked" rectangulations and drawings of planar maps. These observations arise from viewing a generic rectangulation as topologically equivalent to a sphere
Reconfiguration of plane trees in convex geometric graphs
A non-crossing spanning tree of a set of points in the plane is a spanning
tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that
there always exists a flip sequence of length at most between any pair
of non-crossing spanning trees (where denotes the number of points).
Hernando et al. proved that the length of a minimal flip sequence can be of
length at least . Two recent results of Aichholzer et al. and
Bousquet et al. improved the Avis and Fukuda upper bound by proving that there
always exists a flip sequence of length respectively at most and
. We improve the upper bound by a linear factor for the first
time in 25 years by proving that there always exists a flip sequence between
any pair of non-crossing spanning trees of length at most where
. Our result is actually stronger since we prove that, for any
two trees , there exists a flip sequence from to of length
at most . We also improve the best lower bound in terms
of the symmetric difference by proving that there exists a pair of trees
such that a minimal flip sequence has length , improving the lower bound of Hernando et al. by considering the
symmetric difference instead of the number of vertices. We generalize this
lower bound construction to non-crossing flips (where we close the gap between
upper and lower bounds) and rotations
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum