  The future of main memory appears to lie in the direction of new technologies
that provide strong capacity-to-performance ratios, but have write operations
that are much more expensive than reads in terms of latency, bandwidth, and
energy. Motivated by this trend, we propose sequential and parallel algorithms
to solve graph connectivity problems using significantly fewer writes than
conventional algorithms. Our primary algorithmic tool is the construction of an
$o(n)$-sized "implicit decomposition" of a bounded-degree graph $G$ on $n$
nodes, which combined with read-only access to $G$ enables fast answers to
connectivity and biconnectivity queries on $G$. The construction breaks the
linear-write "barrier", resulting in costs that are asymptotically lower than
conventional algorithms while adding only a modest cost to querying time. For
general non-sparse graphs on $m$ edges, we also provide the first $o(m)$ writes
and $O(m)$ operations parallel algorithms for connectivity and biconnectivity.
These algorithms provide insight into how applications can efficiently process
computations on large graphs in systems with read-write asymmetry.
