Hello.

With hash of hash, for looking ups.

#!/usr/bin/perl
use strict; use warnings; use Data::Dumper; 
my @a=(1,1,2,3,4,4,8,8);  
my @b=(3,4,3,5,6,7,9,10);
my $hoh={};

for ( 0 .. $#a){
    $hoh->{$a[$_]}->{$b[$_]}++;
    $hoh->{$b[$_]}->{$a[$_]}++;
}
sub recursive {
    my ($hoh, $key, $path, @vals) =@_;
    for ( grep { $_ ne $key } @vals){
        next if exists($path->{$_});
        $hoh->{$key}->{$_}++;
        $path->{$_}++;
        recursive($hoh, $key, $path, keys %{$hoh->{$_}} );
    }
}
for (sort {$a <=> $b}keys %$hoh){
    recursive($hoh, $_, {}, keys %{$hoh->{$_}} );   
}

#print Dumper $hoh;
for (sort {$a <=> $b}keys %$hoh){
    print "$_=" , join(",", keys %{$hoh->{$_}}), "\n";
}

__DATA__
this prints ...
1=6,4,3,7,2,5
2=6,4,1,3,7,5
3=6,4,1,7,2,5
4=6,1,3,7,2,5
5=6,1,4,3,7,2
6=1,4,3,7,2,5
7=6,1,4,3,2,5
8=10,9
9=8,10
10=8,9
[download]

And as BillKSmith points out, if you take this as a graph,

#!/usr/bin/perl
#with graph
use strict; use warnings; use Data::Dumper;
use Graph;
my $g =Graph::Undirected->new;

my @a=(1,1,2,3,4,4,8,8);
my @b=(3,4,3,5,6,7,9,10); 
    
for ( 0 .. $#a){
    $g->add_edge($a[$_], $b[$_]);
}   
print Dumper $g->connected_components;
__DATA__
this prints ...$VAR1 = [
          9,
          8,
          10
        ];
$VAR2 = [
          6,
          4,
          1,
          3,
          2,
          5,
          7
        ];
[download]

regards.