How about just clearing @big before you make the top the top-level call to walk(). Is that what you want to do?

our @big; # "my" will also work here

for my $r (@array) {
  @big = ();
  walk($r);
  # @big lists every node visited by walk($r)
  ...do something with @big...
}

sub walk {
  my $i = shift;
  push(@big, $i);
  for (...) { walk($_) }
}
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