  Compact and I/O-efficient data representations play an important role in
efficient algorithm design, as memory bandwidth and latency can present a
significant performance bottleneck, slowing the computation by orders of
magnitude. While this problem is very well explored in e.g. uniform numerical
data processing, structural data applications (e.g. on huge graphs) require
different algorithm-dependent approaches. Separable graph classes (i.e. graph
classes with balanced separators of size $\mathcal{O}(n^c)$ with $c < 1$)
include planar graphs, bounded genus graphs, and minor-free graphs.
  In this article we present two generalizations of the separator theorem, to
partitions with small regions only on average and to weighted graphs. Then we
propose I/O-efficient succinct representation and memory layout for random
walks in(weighted) separable graphs in the pointer machine model, including an
efficient algorithm to compute them. Finally, we present a worst-case
I/O-optimal tree layout algorithm for root-leaf path traversal, show an
additive (+1)-approximation of optimal compact layout and contrast this with
NP-completeness proof of finding an optimal compact layout.
