I think your guess may be right. I changed it to have:

    my ($do_search1, $do_search2);
    $do_search2 =  sub {

      my ($node) = @_;
      $sub->($node);
      $explored->{$node->{_id}}++;
      foreach my $link (@{$node->{_outlinks}}) {
          die "too deep" unless ($explored->{$link->{_id}});
      }
    };
    $do_search1 =  sub {

      my ($node) = @_;
      $sub->($node);
      $explored->{$node->{_id}}++;
      foreach my $link (@{$node->{_outlinks}}) {
        $do_search2->($link->{_to}) unless ($explored->{$link->{_id}})
+;
      }
    };
    $do_search =  sub {

      my ($node) = @_;
      $sub->($node);
      $explored->{$node->{_id}}++;
      foreach my $link (@{$node->{_outlinks}}) {
        $do_search1->($link->{_to}) unless ($explored->{$link->{_id}})
+;
      }
    };
[download]

so no recursive calls were done, and got:

BUILDING GRAPH
Neighbors
  Node=HASH(0xa0920e8) ID 1
  Node=HASH(0xa092160) ID 3
  Node=HASH(0xa092124) ID 2
DESTROYING Network=HASH(0xa0420ac)
DESTROYING Link=HASH(0xa0921b4) 2
DESTROYING Link=HASH(0xa092184) 1
DESTROYING Link=HASH(0xa0921e4) 3
DESTROYING Link=HASH(0xa092214) 4
DESTROYING Node=HASH(0xa092160) 3
DESTROYING Node=HASH(0xa0920e8) 1
DESTROYING Node=HASH(0xa092124) 2
SHOULD BE THE LAST THING PRINTED, HOWEVER ...
[download]

Can you simplify it so you just manually bless a few references into "main" and perlbug it?