Is2EdgeTransitive performance update (#739)

* Change to GAP stable-4.14 in CI

* Add HasPseudoSimilarVertices no docs

* Fixed syntax

* fixed syntax

* Add HasPSVs

* Is2EdgeTransitive overhaul using Orbit-Stabiliser Theorem

* Added support for digraphs with loops

* Linting

* More linting

* Edited Documentation

* Edited Documentation again

* Once more rewritten Is2EdgeTransitive, this time to avoid looping through all triples of vertices, and instead checks center vertices of 2-edges

* lint

* Added more comments to prop.gi

* Fix trailing whitespace

* Added IsTwoEdgeTransitive as synonym for Is2EdgeTransitive

* lint

* Removed spurious files and added what happens when the argument of Is2EdgeTransitive has multiple edges to doc

* fixed typo in doc

Author: frankiegillis

doc/prop.xml

@@ -1589,31 +1589,31 @@ false]]></Example>
 </ManSection>
 <#/GAPDoc>
 
-<#GAPDoc Label="IsTwoEdgeTransitive">
+<#GAPDoc Label="Is2EdgeTransitive">
 <ManSection>
-  <Prop Name="IsTwoEdgeTransitive" Arg="digraph"/>
+  <Prop Name="Is2EdgeTransitive" Arg="digraph"/>
   <Returns><K>true</K> or <K>false</K>.</Returns>
   <Description>
-    If <A>digraph</A> is a digraph without multiple edges, then <C>IsTwoEdgeTransitive</C>
+    If <A>digraph</A> is a digraph without multiple edges, then <C>Is2EdgeTransitive</C>
     returns <K>true</K> if <A>digraph</A> is 2-edge transitive, and <K>false</K>
-    otherwise. A digraph is <E>2-edge transitive</E> if its automorphism group
-    acts transitively on its 2-edges via the action
-    <Ref Func="OnTuplesTuples" BookName="ref"/>.
+    otherwise. If <A>digraph</A> has multiple edges, then <C>Is2EdgeTransitive</C> returns an error.
 
-    A <E>2-edge</E> of a digraph is a pair of its edges, such that the range of the
-    first edge is equal to the source of the second edge.
+    A digraph is <E>2-edge transitive</E> if its automorphism group
+    acts transitively on 2-edges via the action
+    <Ref Func="OnTuples" BookName="ref"/>. A <E>2-edge</E> in a digraph is a triple (u, v, w) of distinct vertices
+    such that (u, v) and (v, w) are edges.
     <P/>
 
     &MUTABLE_RECOMPUTED_PROP;
 
     <Example><![CDATA[
-gap> IsTwoEdgeTransitive(CompleteDigraph(4));
+gap> Is2EdgeTransitive(CompleteDigraph(4));
 true
-gap> IsTwoEdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 1], [3, 4]]))
+gap> Is2EdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 4]]));
 false
-gap> IsTwoEdgeTransitive(CycleDigraph(5));
+gap> Is2EdgeTransitive(CycleDigraph(5));
 true
-gap> IsTwoEdgeTransitive(Digraph([[2], [3, 3, 3], []]));
+gap> Is2EdgeTransitive(Digraph([[2], [3, 3, 3], []]));
 Error, the argument <D> must be a digraph with no multiple edges,
 ]]></Example>
   </Description>

doc/z-chap5.xml

@@ -78,6 +78,7 @@
     <#Include Label="IsDigraphCore">
     <#Include Label="IsEdgeTransitive">
     <#Include Label="IsVertexTransitive">
+    <#Include Label="Is2EdgeTransitive">
   </Section>
 
 

gap/prop.gd

@@ -53,7 +53,7 @@ DeclareProperty("IsMeetSemilatticeDigraph", IsDigraph);
 DeclareProperty("IsPermutationDigraph", IsDigraph);
 DeclareProperty("IsDistributiveLatticeDigraph", IsDigraph);
 DeclareProperty("IsModularLatticeDigraph", IsDigraph);
-DeclareProperty("IsTwoEdgeTransitive", IsDigraph);
+DeclareProperty("Is2EdgeTransitive", IsDigraph);
 DeclareSynonymAttr("IsLatticeDigraph",
                    IsMeetSemilatticeDigraph and IsJoinSemilatticeDigraph);
 DeclareSynonymAttr("IsPreorderDigraph",
@@ -70,6 +70,7 @@ DeclareSynonymAttr("IsEquivalenceDigraph",
 DeclareSynonymAttr("IsAntiSymmetricDigraph", IsAntisymmetricDigraph);
 DeclareSynonymAttr("IsNullDigraph", IsEmptyDigraph);
 DeclareSynonymAttr("IsQuasiorderDigraph", IsPreorderDigraph);
+DeclareSynonymAttr("IsTwoEdgeTransitive", Is2EdgeTransitive);
 
 DeclareOperation("DIGRAPHS_IsJoinSemilatticeAndJoinTable", [IsDigraph]);
 DeclareOperation("DIGRAPHS_IsMeetSemilatticeAndMeetTable", [IsDigraph]);

gap/prop.gi

@@ -693,25 +693,94 @@ function(D)
   return LatticeDigraphEmbedding(N5, D) = fail;
 end);
 
-InstallMethod(IsTwoEdgeTransitive,
+InstallMethod(Is2EdgeTransitive,
 "for a digraph without multiple edges",
 [IsDigraph],
 function(D)
-  local twoEdges;
-
+  local Aut, O, I, Centers, Count, In, Out, u;
   if IsMultiDigraph(D) then
     ErrorNoReturn("the argument <D> must be a digraph with no multiple",
                   " edges,");
   fi;
 
-  twoEdges := Filtered(Cartesian(DigraphEdges(D), DigraphEdges(D)),
-                       pair -> pair[1][2] = pair[2][1]
-                               and pair[1][1] <> pair[2][2]);
+  Aut := AutomorphismGroup(D);
+  D := DigraphRemoveLoops(D);
+  O := D!.OutNeighbours;
+  I := InNeighbours(D);
+  # The list Centers will store all those vertices which lie at the
+  # center of a 2-edge.
+
+  Centers := [];
+
+  for u in [1 .. Length(O)] do
+    if Length(O[u]) > 0 and Length(I[u]) > 0 then
+      # If u has precisely one in neighbour and out neighbour,
+      # we must check these are not the same vertex as then there
+      # would be no 2-edge centered at u.
+
+      if Length(O[u]) = 1 and Length(I[u]) = 1 then
+        if O[u][1] = I[u][1] then
+          continue;
+        fi;
+      fi;
+      if not IsBound(Out) then
+        Out := Length(O[u]);
+        In := Length(I[u]);
+      fi;
+      # For D to be 2-edge transitive, it must be transitive
+      # on 2-edge centers, so all 2-edge centers must have the
+      # same in-degree and same out-degree.
 
-  if Length(twoEdges) = 0 then
+      if Out <> Length(O[u]) or In <> Length(I[u]) then
+        return false;
+      fi;
+      Add(Centers, u);
+    fi;
+  od;
+  # If Centers is empty, D has no 2-edges so is vacuously 2-edge
+  # transtive.
+
+  if Length(Centers) = 0 then
     return true;
-  else
-    return OrbitLength(AutomorphismGroup(D), twoEdges[1], OnTuplesTuples)
-         = Length(twoEdges);
   fi;
+  # Find the number of 2-cycles at any center. We will have to subtract
+  # these from the total number of 2-edges as 2-cycles are not classed
+  # as 2-edges.
+
+  Count := 0;
+  for u in O[Centers[1]] do
+    if Centers[1] in O[u] then
+      Count := Count + 1;
+    fi;
+  od;
+
+  # Find a 2-edge and check if its orbit length equals the number of 2-edges.
+  # By this point, we know that D is likely a highly symmetric digraph,
+  # since all 2-edge centers share a common in and out degree
+  # (This is by no means a guarantee, see Frucht's graph). From testing,
+  # calculating the stabilizer and using the orbit-stabilizer
+  # theorem is usually must faster in this case, so we instead determine
+  # the stabilizer of a 2-edge.
+
+  for u in I[Centers[1]] do
+    if Position(O[Centers[1]], u) = 1 then
+      if Length(O[Centers[1]]) = 1 then
+        continue;
+      else
+        return (In * Out - Count) * Length(Centers) =
+               Order(Aut) / Order(Stabilizer(Aut,
+                                             [u,
+                                              Centers[1],
+                                              O[Centers[1]][2]],
+                                             OnTuples));
+      fi;
+    else
+      return (In * Out - Count) * Length(Centers) =
+             Order(Aut) / Order(Stabilizer(Aut,
+                                           [u,
+                                            Centers[1],
+                                            O[Centers[1]][1]],
+                                           OnTuples));
+    fi;
+  od;
 end);

tst/standard/prop.tst

@@ -1738,22 +1738,22 @@ true
 gap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));
 Error, the argument <D> must be a digraph with no multiple edges,
 
-# IsTwoEdgeTransitive
-gap> IsTwoEdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 1]]));
+# Is2EdgeTransitive
+gap> Is2EdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 1]]));
 true
-gap> IsTwoEdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 1], [3, 4]]));
+gap> Is2EdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 1], [3, 4]]));
 false
-gap> IsTwoEdgeTransitive(CompleteDigraph(4));
+gap> Is2EdgeTransitive(CompleteDigraph(4));
 true
-gap> IsTwoEdgeTransitive(CycleDigraph(100));
+gap> Is2EdgeTransitive(CycleDigraph(100));
 true
-gap> IsTwoEdgeTransitive(CompleteBipartiteDigraph(11, 23));
+gap> Is2EdgeTransitive(CompleteBipartiteDigraph(11, 23));
 false
-gap> IsTwoEdgeTransitive(DigraphByEdges([[1, 2]]));
+gap> Is2EdgeTransitive(DigraphByEdges([[1, 2]]));
 true
-gap> IsTwoEdgeTransitive(DigraphByEdges([]));
+gap> Is2EdgeTransitive(DigraphByEdges([]));
 true
-gap> IsTwoEdgeTransitive(Digraph([[2], [3, 3, 3], []]));
+gap> Is2EdgeTransitive(Digraph([[2], [3, 3, 3], []]));
 Error, the argument <D> must be a digraph with no multiple edges,
 
 # DigraphHasNoVertices and DigraphHasAVertex

tst/testinstall.tst

@@ -440,8 +440,8 @@ rec( distances := [ [ 0, 5 ], [ 5, 0 ] ],
 gap> EdgeWeightedDigraphShortestPath(d, 1, 2);
 [ [ 1, 2 ], [ 1 ] ]
 
-# IsTwoEdgeTransitive
-gap> IsTwoEdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 1]]));
+# Is2EdgeTransitive
+gap> Is2EdgeTransitive(DigraphByEdges([[1, 2], [2, 3], [3, 1]]));
 true
 
 # Issue 617: bug in DigraphRemoveEdge, wasn't removing edge labels