diff --git a/doc/oper.xml b/doc/oper.xml
index 1bd1fc966..d60a391aa 100644
--- a/doc/oper.xml
+++ b/doc/oper.xml
@@ -1353,23 +1353,29 @@ false
 <#GAPDoc Label="VerticesReachableFrom">
 <ManSection>
   <Oper Name="VerticesReachableFrom" Arg="digraph, root"/>
+  <Oper Name="VerticesReachableFrom" Arg="digraph, list"/>
   <Returns>A list.</Returns>
   <Description>
-    This operation returns a list of the vertices  <A>v</A>, for which there exists
-    a non-trivial directed walk from vertex <A>root</A> to vertex <A>v</A> in the digraph
+    This operation returns a list of the vertices <C>v</C>, for which there
+    exists a non-trivial directed walk from the vertex <A>root</A>, or any of
+    the list of vertices <A>list</A>, to vertex <C>v</C> in the digraph
     <A>digraph</A>. See Section <Ref Subsect="Definitions" Style="Number" />
     for the definition of a non-trivial directed walk.
     <P/>
 
-    The method for <C>VerticesReachableFrom</C> has worst case complexity of <M>O(m +
-      n)</M> where <M>m</M> is the number of edges and <M>n</M> the number of
-    vertices in <A>digraph</A>.
+    The method for <C>VerticesReachableFrom</C> has worst case complexity of
+    <M>O(m + n)</M> where <M>m</M> is the number of edges and <M>n</M> the
+    number of vertices in <A>digraph</A>.
+    <P/>
+
+    This function returns an error if <A>root</A>, or any vertex in <A>list</A>,
+    is not a vertices of <A>digraph</A>.
 
 <Example><![CDATA[
 gap> D := CompleteDigraph(5);
 <immutable complete digraph with 5 vertices>
 gap> VerticesReachableFrom(D, 1);
-[ 2, 1, 3, 4, 5 ]
+[ 1, 2, 3, 4, 5 ]
 gap> VerticesReachableFrom(D, 3);
 [ 1, 2, 3, 4, 5 ]
 gap> D := EmptyDigraph(5);
@@ -1378,12 +1384,16 @@ gap> VerticesReachableFrom(D, 1);
 [  ]
 gap> VerticesReachableFrom(D, 3);
 [  ]
+gap> VerticesReachableFrom(D, [1, 2, 3, 4]);
+[  ]
+gap> VerticesReachableFrom(D, [3, 4, 5]);
+[  ]
 gap> D := CycleDigraph(4);
 <immutable cycle digraph with 4 vertices>
 gap> VerticesReachableFrom(D, 1);
-[ 2, 3, 4, 1 ]
+[ 1, 2, 3, 4 ]
 gap> VerticesReachableFrom(D, 3);
-[ 4, 1, 2, 3 ]
+[ 1, 2, 3, 4 ]
 gap> D := ChainDigraph(5);
 <immutable chain digraph with 5 vertices>
 gap> VerticesReachableFrom(D, 1);
@@ -1392,12 +1402,13 @@ gap> VerticesReachableFrom(D, 3);
 [ 4, 5 ]
 gap> VerticesReachableFrom(D, 5);
 [  ]
+gap> VerticesReachableFrom(D, [3, 4]);
+[ 4, 5 ]
 ]]></Example>
   </Description>
 </ManSection>
 <#/GAPDoc>
 
-
 <#GAPDoc Label="DigraphPath">
 <ManSection>
   <Oper Name="DigraphPath" Arg="digraph, u, v"/>
@@ -2257,3 +2268,30 @@ true
   </Description>
 </ManSection>
 <#/GAPDoc>
+
+<#GAPDoc Label="IsOrderIdeal">
+<ManSection>
+  <Oper Name="IsOrderIdeal" Arg="D, subset" Label="for a digraph and list"/>
+  <Returns><K>true</K> or <K>false</K>.</Returns>
+  <Description>
+    This function returns <K>true</K> if the specified subset is "downwards" closed, i.e. contains every vertex less than the given vertices in the order defined by <A>D</A>. 
+    The function can only be used on digraphs satisfying <Ref Prop="IsPartialOrderDigraph"/> and will throw an error if passed a digraph that is not a partial order digraph.
+
+    <Example><![CDATA[
+gap> D1 := Digraph([[1, 3], [2, 3], [3]]);
+<immutable digraph with 3 vertices, 5 edges>
+<immutable digraph with 3 vertices, 5 edges>
+gap> IsOrderIdeal(D, [1, 2, 3]);
+true
+gap> D2 := DigraphDisjointUnion(D, D);
+<immutable digraph with 6 vertices, 10 edges>
+gap> IsOrderIdeal(D2, [1, 2, 3]);
+true
+gap> IsOrderIdeal(D2, [4, 5, 6]);
+true
+gap> IsOrderIdeal(D2, [1, 5, 6]);
+false
+]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
diff --git a/gap/oper.gd b/gap/oper.gd
index 5c1ea6046..5594ea8c8 100644
--- a/gap/oper.gd
+++ b/gap/oper.gd
@@ -137,6 +137,8 @@ DeclareOperation("DigraphShortestDistance", [IsDigraph, IsList]);
 DeclareOperation("DigraphShortestPath", [IsDigraph, IsPosInt, IsPosInt]);
 DeclareOperation("DigraphShortestPathSpanningTree", [IsDigraph, IsPosInt]);
 DeclareOperation("VerticesReachableFrom", [IsDigraph, IsPosInt]);
+DeclareOperation("VerticesReachableFrom", [IsDigraph, IsList]);
+DeclareOperation("IsOrderIdeal", [IsDigraph, IsList]);
 DeclareOperation("Dominators", [IsDigraph, IsPosInt]);
 DeclareOperation("DominatorTree", [IsDigraph, IsPosInt]);
 
diff --git a/gap/oper.gi b/gap/oper.gi
index 5186a7d3f..4cb26c81b 100644
--- a/gap/oper.gi
+++ b/gap/oper.gi
@@ -2005,42 +2005,75 @@ end);
 InstallMethod(VerticesReachableFrom, "for a digraph and a vertex",
 [IsDigraph, IsPosInt],
 function(D, root)
-  local N, index, current, succ, visited, prev, n, i, parent,
-  have_visited_root;
+  local N;
   N := DigraphNrVertices(D);
+
   if 0 = root or root > N then
     ErrorNoReturn("the 2nd argument (root) is not a vertex of the 1st ",
                   "argument (a digraph)");
   fi;
-  index := ListWithIdenticalEntries(N, 0);
-  have_visited_root := false;
-  index[root] := 1;
-  current := root;
-  succ := OutNeighbours(D);
-  visited := [];
-  parent := [];
-  parent[root] := fail;
-  repeat
-    prev := current;
-    for i in [index[current] .. Length(succ[current])] do
-      n := succ[current][i];
-      if n = root and not have_visited_root then
-         Add(visited, root);
-         have_visited_root := true;
-      elif index[n] = 0 then
-        Add(visited, n);
-          parent[n] := current;
-          index[current] := i + 1;
-          current := n;
-          index[current] := 1;
-          break;
+
+  return VerticesReachableFrom(D, [root]);
+end);
+
+InstallMethod(VerticesReachableFrom, "for a digraph and a list of vertices",
+[IsDigraph, IsList],
+function(D, roots)
+  local N, index, visited, queue_tail, queue,
+  root, element, neighbour, graph_out_neighbors, node_neighbours;
+
+  N := DigraphNrVertices(D);
+
+  for root in roots do
+    if not IsPosInt(N) or 0 = root or root > N then
+      ErrorNoReturn("an element of the 2nd argument ",
+                    "(roots) is not a vertex of the 1st ",
+                    "argument (a digraph)");
+    fi;
+  od;
+
+  visited := BlistList([1 .. N], []);
+
+  graph_out_neighbors := OutNeighbors(D);
+  queue := EmptyPlist(N);
+  Append(queue, roots);
+
+  queue_tail := Length(roots);
+
+  index := 1;
+  while IsBound(queue[index]) do
+    element := queue[index];
+    node_neighbours := graph_out_neighbors[element];
+    for neighbour in node_neighbours do
+      if not visited[neighbour] then;
+        visited[neighbour] := true;
+        queue_tail := queue_tail + 1;
+        queue[queue_tail] := neighbour;
       fi;
     od;
-    if prev = current then
-      current := parent[current];
-    fi;
-  until current = fail;
-  return visited;
+    index := index + 1;
+  od;
+
+  return ListBlist([1 .. N], visited);
+end);
+
+InstallMethod(IsOrderIdeal, "for a digraph and a list of vertices",
+[IsDigraph, IsList],
+# Check if digraph represents a partial order
+function(D, roots)
+  local reachable_vertices, vertex_in_subset, N;
+  if not IsPartialOrderDigraph(D) then
+    ErrorNoReturn(
+      "the 1st argument (a digraph) must be a partial order digraph");
+  fi;
+
+  N := DigraphNrVertices(D);
+  vertex_in_subset := BlistList([1 .. N], roots);
+  reachable_vertices := VerticesReachableFrom(D, roots);
+
+  return Length(reachable_vertices) = Length(roots)
+    and ForAll(reachable_vertices, x -> vertex_in_subset[x]);
+
 end);
 
 InstallMethod(DominatorTree, "for a digraph and a vertex",
diff --git a/tst/standard/oper.tst b/tst/standard/oper.tst
index 80086203b..476cb9f66 100644
--- a/tst/standard/oper.tst
+++ b/tst/standard/oper.tst
@@ -1942,7 +1942,7 @@ gap> DigraphShortestPath(gr, 12, 5014);
 gap> D := CompleteDigraph(5);
 <immutable complete digraph with 5 vertices>
 gap> VerticesReachableFrom(D, 1);
-[ 2, 1, 3, 4, 5 ]
+[ 1, 2, 3, 4, 5 ]
 gap> VerticesReachableFrom(D, 3);
 [ 1, 2, 3, 4, 5 ]
 gap> D := EmptyDigraph(5);
@@ -1951,12 +1951,14 @@ gap> VerticesReachableFrom(D, 1);
 [  ]
 gap> VerticesReachableFrom(D, 3);
 [  ]
+gap> VerticesReachableFrom(D, 6);
+Error, the 2nd argument (root) is not a vertex of the 1st argument (a digraph)
 gap> D := CycleDigraph(4);
 <immutable cycle digraph with 4 vertices>
 gap> VerticesReachableFrom(D, 1);
-[ 2, 3, 4, 1 ]
+[ 1, 2, 3, 4 ]
 gap> VerticesReachableFrom(D, 3);
-[ 4, 1, 2, 3 ]
+[ 1, 2, 3, 4 ]
 gap> D := ChainDigraph(5);
 <immutable chain digraph with 5 vertices>
 gap> VerticesReachableFrom(D, 1);
@@ -1968,29 +1970,29 @@ gap> VerticesReachableFrom(D, 5);
 gap> D := Digraph([[2, 3, 5], [1, 6], [4, 6, 7], [7, 8], [4], [], [8, 6], []]);
 <immutable digraph with 8 vertices, 13 edges>
 gap> VerticesReachableFrom(D, 1);
-[ 2, 1, 6, 3, 4, 7, 8, 5 ]
+[ 1, 2, 3, 4, 5, 6, 7, 8 ]
 gap> VerticesReachableFrom(D, 2);
-[ 1, 2, 3, 4, 7, 8, 6, 5 ]
+[ 1, 2, 3, 4, 5, 6, 7, 8 ]
 gap> VerticesReachableFrom(D, 3);
-[ 4, 7, 8, 6 ]
+[ 4, 6, 7, 8 ]
 gap> VerticesReachableFrom(D, 4);
-[ 7, 8, 6 ]
+[ 6, 7, 8 ]
 gap> VerticesReachableFrom(D, 5);
-[ 4, 7, 8, 6 ]
+[ 4, 6, 7, 8 ]
 gap> VerticesReachableFrom(D, 6);
 [  ]
 gap> VerticesReachableFrom(D, 7);
-[ 8, 6 ]
+[ 6, 8 ]
 gap> VerticesReachableFrom(D, 8);
 [  ]
 gap> D := Digraph([[1, 2, 3], [4], [1, 5], [], [2]]);
 <immutable digraph with 5 vertices, 7 edges>
 gap> VerticesReachableFrom(D, 1);
-[ 1, 2, 4, 3, 5 ]
+[ 1, 2, 3, 4, 5 ]
 gap> VerticesReachableFrom(D, 2);
 [ 4 ]
 gap> VerticesReachableFrom(D, 3);
-[ 1, 2, 4, 3, 5 ]
+[ 1, 2, 3, 4, 5 ]
 gap> VerticesReachableFrom(D, 4);
 [  ]
 gap> VerticesReachableFrom(D, 5);
@@ -1998,11 +2000,11 @@ gap> VerticesReachableFrom(D, 5);
 gap> D := Digraph(IsMutableDigraph, [[1, 2, 3], [4], [1, 5], [], [2]]);
 <mutable digraph with 5 vertices, 7 edges>
 gap> VerticesReachableFrom(D, 1);
-[ 1, 2, 4, 3, 5 ]
+[ 1, 2, 3, 4, 5 ]
 gap> VerticesReachableFrom(D, 2);
 [ 4 ]
 gap> VerticesReachableFrom(D, 3);
-[ 1, 2, 4, 3, 5 ]
+[ 1, 2, 3, 4, 5 ]
 gap> VerticesReachableFrom(D, 4);
 [  ]
 gap> VerticesReachableFrom(D, 5);
@@ -2796,12 +2798,54 @@ gap> D := Digraph([
 gap> path := DigraphPath(D, 5, 5);;
 gap> IsDigraphPath(D, path);
 true
+gap> D1 := CompleteDigraph(5);
+<immutable complete digraph with 5 vertices>
+gap> D2 := CompleteDigraph(10);
+<immutable complete digraph with 10 vertices>
+gap> VerticesReachableFrom(D1, [1]);
+[ 1, 2, 3, 4, 5 ]
+gap> VerticesReachableFrom(D1, [1, 2]);
+[ 1, 2, 3, 4, 5 ]
+gap> VerticesReachableFrom(D2, [1]);
+[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
+gap> VerticesReachableFrom(D2, [1, 11]);
+Error, an element of the 2nd argument (roots) is not a vertex of the 1st argum\
+ent (a digraph)
+gap> D3 := CompleteDigraph(7);
+<immutable complete digraph with 7 vertices>
+gap> D3_edges := [1 .. 7];
+[ 1 .. 7 ]
+gap> for i in D3_edges do
+>      D3 := DigraphRemoveEdge(D3, [1, i]);
+>      D3 := DigraphRemoveEdge(D3, [i, 1]);
+>    od;
+gap> VerticesReachableFrom(D3, [1]);
+[  ]
+gap> TestPartialOrderDigraph := Digraph([[1, 3], [2, 3], [3]]);
+<immutable digraph with 3 vertices, 5 edges>
+gap> IsOrderIdeal(TestPartialOrderDigraph, [1, 2, 3]);
+true
+gap> TestPartialOrderDigraph2 := Digraph([[1, 3], [2, 3], [3]]);
+<immutable digraph with 3 vertices, 5 edges>
+gap> TestUnion := DigraphDisjointUnion(TestPartialOrderDigraph, TestPartialOrderDigraph2);
+<immutable digraph with 6 vertices, 10 edges>
+gap> IsOrderIdeal(TestUnion, [1, 2, 3]);
+true
+gap> IsOrderIdeal(TestUnion, [4, 5, 6]);
+true
+gap> IsOrderIdeal(TestUnion, [1, 5, 6]);
+false
+gap> D := CycleDigraph(5);;
+gap> IsOrderIdeal(D, [1]);
+Error, the 1st argument (a digraph) must be a partial order digraph
 
 #  DIGRAPHS_UnbindVariables
 gap> Unbind(C);
 gap> Unbind(D);
 gap> Unbind(D1);
 gap> Unbind(D2);
+gap> Unbind(D3);
+gap> Unbind(D3_edges);
 gap> Unbind(DD);
 gap> Unbind(G);
 gap> Unbind(G1);
@@ -2854,6 +2898,7 @@ gap> Unbind(tclosure);
 gap> Unbind(u1);
 gap> Unbind(u2);
 gap> Unbind(x);
+gap> Unbind(TestPartialOrderDigraph);
 
 #
 gap> DIGRAPHS_StopTest();