It sounds like you are looking for an edge whose removal disconnects the graph (edge 4,7 in your case). This is called a "cut-edge" or a "bridge". Fortunately, Graph has a simple way to find all the bridges:

use Graph::Undirected;

my $g = Graph::Undirected->new;
$g->add_edges(
  map [ split /,/, $_ ], qw[ 1,2 1,4 2,3 3,4 4,7 7,8 7,10 8,9 9,10 ]
);

my @bridges = $g->bridges;

print "Found bridge(s):\n";
print " - @$_\n" for @bridges;

$g->delete_edge(@$_) for @bridges;

print "Resulting components:\n";
print " - @$_\n" for $g->connected_components;

__END__

Found bridge(s):
 - 4 7
Resulting components:
 - 4 1 2 3
 - 8 9 10 7
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I'm not sure, but the bridges method may not work if the graph is not connected (the biconnectivity method computes the same information and has such a warning in the documentation).

Update: showed how to remove bridges to obtain list of "clusters" that is induced.