Attempt 3:

#! perl -slw
use strict;
use Algorithm::Combinatorics qw[ combinations ];

my $favourite = 1009;
my $n = <>;
my %pairings;
push( @{ $pairings{ $_->[0] } }, $_->[1] ),
push( @{ $pairings{ $_->[1] } }, $_->[0] ),
    while @{ $_ = [split ' ', <> ] };

my @ids = keys %pairings;

my %idPos; @idPos{ @ids } = 0 .. @ids;
my %posId; @posId{ values %idPos } = keys %idPos;

my @masks;
for my $id ( @ids ) {
    $masks[ $idPos{ $id } ] = '';
    vec( $masks[ $idPos{ $id } ], $idPos{ $id }, 1 ) = 1;
    vec( $masks[ $idPos{ $id } ], $idPos{ $_  }, 1 ) = 1
        for @{ $pairings{ $id } };
}

my @hits;
for my $k ( 1 .. $n ) {
    my $iter = combinations( [0 .. $#ids], $k );
    while( my $c = $iter->next ) {
        my $ored = '';
        $ored |= $masks[ $_ ] for @$c;
        my $count = unpack '%32b*', $ored;
        push @hits, [ $k, @$c ] if $count >= @ids;
    }
    last if @hits;
}

if( @hits > 1 ) {
    @hits = sort{ $a->[0] <=> $b->[0] } @hits;
    my $min = $hits[0][0];
    my $i = $#hits;
    --$i while $hits[ $i ][0] > $min;
    $#hits = $i;
    if( @hits > 1 ) {
        my $first = $hits[ 0 ];
        @hits = grep {
            scalar grep{
                $posId{ $_ } == $favourite;
            } @$_[ 0 .. $#{ $_ } ];
        } @hits;
        @hits = @hits >= 1 ? $hits[ 0 ] : $first;
    }
}

print @{ $hits[0] } -1;
print $posId{ $_ } for @{ $hits[0] }[ 1 .. $#{ $hits[ 0 ] } ];
[download]

A run:

C:\test>Bilateral2.pl
20
1001 2000
1002 2000
1009 2000
1011 2010
1012 2010
1013 2010
1021 2020
1022 2020
1023 2020
1031 2030
1032 2030
1033 2030
1041 2040
1042 2040
1043 2040
1100 2000
1100 2010
1100 2020
1100 2030
1100 2040
^Z
5
2020
2030
2000
2010
2040
[download]