diff --git a/gap/cliques.gi b/gap/cliques.gi
index 79dd45834..4275e6433 100644
--- a/gap/cliques.gi
+++ b/gap/cliques.gi
@@ -723,9 +723,9 @@ function(gr, hook, user_param, limit, include, exclude, max, size, reps)
 end);
 
 InstallGlobalFunction(DIGRAPHS_BronKerbosch,
-function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
-  local vtx, grp, invariant_inc, invariant_exc, invariant, adj, exc_inv, start,
-  possible, found_orbits, add, bk, num, x;
+function(gr, hook, param, lim, inc, exc, max, size, reps, inc_var, exc_var)
+  local vtx, invariant_inc, invariant_exc, invariant, adj, exc_inv, start,
+  possible, isolated, grp, found_orbits, add, bk, num, x, gen;
 
   # Arguments must be:
   # gr   - a digraph
@@ -759,7 +759,6 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
 
   gr  := MaximalSymmetricSubdigraphWithoutLoops(gr);
   vtx := DigraphVertices(gr);
-  grp := AutomorphismGroup(gr);
 
   invariant_inc := Length(inc_var) = 0;
   invariant_exc := Length(exc_var) = 0;
@@ -793,6 +792,22 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
     IntersectBlist(possible, adj[x]);
   od;
 
+  isolated := DigraphSources(gr); # sources = sinks = isolated vertices
+  if reps and not IsEmpty(isolated) then
+    # Optimisation for when there are isolated vertices
+    grp := Group(());
+    for gen in GeneratorsOfGroup(AutomorphismGroup(gr)) do
+      # Discard generators which act on the isolated points
+      if not SmallestMovedPoint(gen) in isolated then
+        grp := ClosureGroup(grp, gen);
+      fi;
+    od;
+    SubtractBlist(possible, BlistList(vtx, isolated));
+    possible[isolated[1]] := true;
+  else
+    grp := AutomorphismGroup(gr);
+  fi;
+
   found_orbits := [];
 
   # Function to find the valid cliques of an orbit given an orbit rep
@@ -801,7 +816,7 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
 
     c := ListBlist(vtx, c);
     if reps then # we are only looking for orbit reps, so add the rep
-      hook(user_param, c);
+      hook(param, c);
       num := num + 1;
       return;
     elif not ForAny(found_orbits, x -> c in x) then
@@ -814,7 +829,7 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
                         # are invariant so there is nothing extra to check
         n := Minimum(lim - num, n);
         for c in orb{[1 .. n]} do
-          hook(user_param, c);
+          hook(param, c);
         od;
         num := num + n;
         return;
@@ -827,7 +842,7 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
           i := i + 1;
           c := BlistList(vtx, orb[i]);
           if SizeBlist(IntersectionBlist(exc_var, c)) = 0 then
-            hook(user_param, orb[i]);
+            hook(param, orb[i]);
             num := num + 1;
           fi;
         od;
@@ -838,7 +853,7 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
           i := i + 1;
           c := BlistList(vtx, orb[i]);
           if IsSubsetBlist(c, inc_var) then
-            hook(user_param, orb[i]);
+            hook(param, orb[i]);
             num := num + 1;
           fi;
         od;
@@ -851,7 +866,7 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
           c := BlistList(vtx, orb[i]);
           if SizeBlist(IntersectionBlist(exc_var, c)) = 0
               and IsSubsetBlist(c, inc_var) then
-            hook(user_param, orb[i]);
+            hook(param, orb[i]);
             num := num + 1;
           fi;
         od;
@@ -973,5 +988,5 @@ function(gr, hook, user_param, lim, inc, exc, max, size, reps, inc_var, exc_var)
   num := 0;
 
   bk(start, possible, BlistList(vtx, []), grp, Length(inc));
-  return user_param;
+  return param;
 end);
diff --git a/tst/standard/cliques.tst b/tst/standard/cliques.tst
index cdb5af3ae..feaf268b0 100644
--- a/tst/standard/cliques.tst
+++ b/tst/standard/cliques.tst
@@ -503,6 +503,24 @@ gap> c := DigraphMaximalCliques(gr);;
 gap> Length(c);
 12815
 
+#T# Issue #23: Digraphs with isolated vertices
+gap> gr := DigraphFromSparse6String(":~?@c__EC?_F");
+<digraph with 100 vertices, 6 edges>
+gap> DigraphMaximalCliquesReps(gr);
+[ [ 1 ], [ 2, 3, 5 ] ]
+gap> DigraphMaximalCliques(gr);
+[ [ 1 ], [ 4 ], [ 6 ], [ 7 ], [ 8 ], [ 9 ], [ 10 ], [ 11 ], [ 12 ], [ 13 ], 
+  [ 14 ], [ 15 ], [ 16 ], [ 17 ], [ 18 ], [ 19 ], [ 20 ], [ 21 ], [ 22 ], 
+  [ 23 ], [ 24 ], [ 25 ], [ 26 ], [ 27 ], [ 28 ], [ 29 ], [ 30 ], [ 31 ], 
+  [ 32 ], [ 33 ], [ 34 ], [ 35 ], [ 36 ], [ 37 ], [ 38 ], [ 39 ], [ 40 ], 
+  [ 41 ], [ 42 ], [ 43 ], [ 44 ], [ 45 ], [ 46 ], [ 47 ], [ 48 ], [ 49 ], 
+  [ 50 ], [ 51 ], [ 52 ], [ 53 ], [ 54 ], [ 55 ], [ 56 ], [ 57 ], [ 58 ], 
+  [ 59 ], [ 60 ], [ 61 ], [ 62 ], [ 63 ], [ 64 ], [ 65 ], [ 66 ], [ 67 ], 
+  [ 68 ], [ 69 ], [ 70 ], [ 71 ], [ 72 ], [ 73 ], [ 74 ], [ 75 ], [ 76 ], 
+  [ 77 ], [ 78 ], [ 79 ], [ 80 ], [ 81 ], [ 82 ], [ 83 ], [ 84 ], [ 85 ], 
+  [ 86 ], [ 87 ], [ 88 ], [ 89 ], [ 90 ], [ 91 ], [ 92 ], [ 93 ], [ 94 ], 
+  [ 95 ], [ 96 ], [ 97 ], [ 98 ], [ 99 ], [ 100 ], [ 2, 3, 5 ] ]
+
 #T# DIGRAPHS_UnbindVariables
 gap> Unbind(f);
 gap> Unbind(c);