  Given a large data graph, trimming techniques can reduce the search space by
removing vertices without outgoing edges. One application is to speed up the
parallel decomposition of graphs into strongly connected components (SCC
decomposition), which is a fundamental step for analyzing graphs. We observe
that graph trimming is essentially a kind of arc-consistency problem, and AC-3,
AC-4, and AC-6 are the most relevant arc-consistency algorithms for application
to graph trimming. The existing parallel graph trimming methods require
worst-case $\mathcal O(nm)$ time and worst-case $\mathcal O(n)$ space for
graphs with $n$ vertices and $m$ edges. We call these parallel AC-3-based as
they are much like the AC-3 algorithm. In this work, we propose AC-4-based and
AC-6-based trimming methods. That is, AC-4-based trimming has an improved
worst-case time of $\mathcal O(n+m)$ but requires worst-case space of $\mathcal
O(n+m)$; compared with AC-4-based trimming, AC-6-based has the same worst-case
time of $\mathcal O(n+m)$ but an improved worst-case space of $\mathcal O(n)$.
We parallelize the AC-4-based and AC-6-based algorithms to be suitable for
shared-memory multi-core machines. The algorithms are designed to minimize
synchronization overhead. For these algorithms, we also prove the correctness
and analyze time complexities with the work-depth model. In experiments, we
compare these three parallel trimming algorithms over a variety of real and
synthetic graphs. Specifically, for the maximum number of traversed edges per
worker by using 16 workers, AC-3-based traverses up to 58.3 and 36.5 times more
edges than AC-6-based trimming and AC-4-based trimming, respectively.
