  A classical problem in computational geometry and graph algorithms is: given
a dynamic set S of geometric shapes in the plane, efficiently maintain the
connectivity of the intersection graph of S. Previous papers studied the
setting where, before the updates, the data structure receives some parameter
P. Then, updates could insert and delete disks as long as at all times the
disks have a diameter that lies in a fixed range [1/P, 1]. The state-of-the-art
for storing disks in a dynamic connectivity data structure is a data structure
that uses O(Pn) space and that has amortized O(P log^4 n) expected amortized
update time. Connectivity queries between disks are supported in O( log n /
loglog n) time. The state-of-the-art for Euclidean disks immediately implies a
data structure for connectivity between axis-aligned squares that have their
diameter in the fixed range [1/P, 1], with an improved update time of O(P log^4
n) amortized time.
  We restrict our attention to axis-aligned squares, and study fully-dynamic
square intersection graph connectivity. Our result is fully-adaptive to the
aspect ratio, spending time proportional to the current aspect ratio {\psi}, as
opposed to some previously given maximum P. Our focus on squares allows us to
simplify and streamline the connectivity pipeline from previous work. When $n$
is the number of squares and {\psi} is the aspect ratio after insertion (or
before deletion), our data structure answers connectivity queries in O(log n /
loglog n) time. We can update connectivity information in O({\psi} log^4 n +
log^6 n) amortized time. We also improve space usage from O(P n log n) to O(n
log^3 n log {\psi}) -- while generalizing to a fully-adaptive aspect ratio --
which yields a space usage that is near-linear in n for any polynomially
bounded {\psi}.
