This was fun :)

First part is basically the same as Re^3: Sub set where all are connected and finds all the cliques.

Last part looks for duplicated nodes, removes one of them from a clique such that the score is higher.

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11109069 # clique
use warnings;
use List::Util qw( uniq sum );

my $edges = <<END =~ s/.*Relationship.*\n//gr;
Person-1  Person-2  Relationship-strength
A         B         92
A         C         7
B         C         2
C         D         88
END

$edges =~ s/^(\w+)\s+(\w+)\b/join ' ', sort $1, $2/gem;  # fix order
print "$edges\n";
my %edges = map +( $_ => qr/(*FAIL)/ ),
  map s/\s+/,/r, $edges =~ /^\w+\s+\w+/gm;
my %cliques;
my %seen;

find( uniq sort $edges =~ /^(\w+)\s+(\w+)\b/gm );    # start with ever
+y node

sub find
  {
  $seen{ my $set = "@_" }++ and return;
  if( my @out = $set =~ /\b(\w+)\b.+\b(\w+)\b(??{ $edges{"$1,$2"} || "
+" })/ )
    {
    for my $node ( @out ) # pair of unconnected nodes, try without eac
+h one
      {
      @_ > 1 and find( grep $_ ne $node, @_ );
      }
    }
  else
    {
    $cliques{ $set }++;                              # it is fully con
+nected
    }
  }

my $uniquecliques = '';
for ( sort { length $b <=> length $a } sort +uniq # ignore subsets
  keys %cliques, map tr/,/ /r, keys %edges )
  {
  my $pattern =  " $_ " =~ s/\w+/\\b$&\\b/gr =~ s/ /.*?/gr;
  $uniquecliques =~ /^$pattern$/m or $uniquecliques .= "$_\n";
  }

print "cliques:\n\n$uniquecliques";

my %scorecache;
sub scoreclique
  {
  $scorecache{ $_[0] } //= do
    {
    my $score = 0;
    "$_[0]:\n$edges" =~ /\b(\w+)\b.*\b(\w+)\b.*:.*^\1\s+\2\s+(\d+)\b(?
+{
      $score += $3 })(*FAIL)/ms;
    $score;
    }
  }

my @cliques = split /\n/, $uniquecliques;
my %count;
$count{$_}++ for split ' ', $uniquecliques;
for my $node ( map +($_) x --$count{$_}, keys %count ) # for all dups
  {
  my ($from, $to) = grep $cliques[$_] =~ /\b$node\b/, 0 .. $#cliques;
  my $fromscore = scoreclique($cliques[$from]) +
    scoreclique($cliques[$to] =~ s/\b$node\b//r);
  my $toscore = scoreclique($cliques[$to]) +
    scoreclique($cliques[$from] =~ s/\b$node\b//r);
  my $mod = $fromscore < $toscore ? $from : $to; # favor highest score
  $cliques[$mod] = join ' ', grep $_ ne $node, split ' ', $cliques[$mo
+d];
  }

my $total = sum map scoreclique($_), @cliques;
print "\nanswer($total):\n\n";
print "$_\n" for sort grep length, @cliques;

__END__
[download]

Outputs:

A B         92
A C         7
B C         2
C D         88

cliques:

A B C
C D

answer(180):

A B
C D
[download]

It passes all the test cases that have been provided :)

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