This is a somewhat tricky problem because later data may require that separate groups that were formed by earlier data need to be merged. Where the code is a little tricky it's worth being a little long winded, use explicit identifier names and comment each case. Consider:

use strict;
use List::Util;

my %nodes;
my @groups;

while (<DATA>) {
    chomp;
    my ($node1, $node2) = split;

    if (! exists $nodes{$node1} && ! exists $nodes{$node2}) {
        # New group
        $nodes{$node1} = @groups;
        $nodes{$node2} = @groups;
        push @groups, [$node1, $node2];
        next;
    }

    if (! exists $nodes{$node1}) {
        # node1 is part of node2's group
        push @{$groups[$nodes{$node2}]}, $node1;
        $nodes{$node1} = $nodes{$node2};
    }

    if (! exists $nodes{$node2}) {
        # node2 is part of node1's group
        push @{$groups[$nodes{$node1}]}, $node2;
        $nodes{$node2} = $nodes{$node1};
    }

    next # Already met this pairing
        if $nodes{$node1} == $nodes{$node2};

    # node1 and node2 are in different groups. Merge the groups
    my ($group, $nulGroup) = ($nodes{$node1}, $nodes{$node2});

    push @{$groups[$group]}, @{$groups[$nulGroup]};
    $nodes{$_} = $group for @{$groups[$nulGroup]};
    $groups[$nulGroup] = undef;
}

@groups = grep {defined} @groups;

for my $group (0 .. $#groups) {
    print 'Group', $group + 1, ': ', join (', ', @{$groups[$group]}), 
+"\n";
}


__DATA__
Contig1  Contig2
Contig1  Contig3
Contig2  Contig1
Contig2  Contig3
Contig3  Contig1
Contig3  Contig2
Contig3  Contig4
Contig4  Contig3
Contig4  Contig5
Contig6  Contig7
Contig7  Contig6
Contig8  Contig9
Contig9  Contig10
Contig10 Contig8
Contig10 Contig11
Contig11 Contig10
C12 Contig11
C12 Contig5
[download]

Prints:

Group1: Contig6, Contig7
Group2: Contig8, Contig9, Contig10, Contig11, C12, Contig1, Contig2, C
+ontig3, Contig4, Contig5
[download]