Add ModularProduct

doc/oper.xml

@@ -1740,3 +1740,45 @@ gap> DigraphDijkstra(D, 1, 2);
 </Description>
 </ManSection>
 <#/GAPDoc>
+
+<#GAPDoc Label="ModularProduct">
+<ManSection>
+  <Oper Name="ModularProduct" Arg="D1, D2"/>
+  <Returns>A digraph.</Returns>
+  <Description>
+  If <A>D1</A> and <A>D2</A> are digraphs without multiple edges, then
+  <C>ModularProduct</C> calculates the <E>modular product digraph</E>
+  (<C>MPD</C>) of <A>D1</A>and <A>D2</A>.
+
+  <C>MPD</C> has vertex set <C>V1 x V2</C> where <C>V1</C> is the vertex set of
+  <A>D1</A> and <C>V2</C> is the vertex set of <A>D2</A> (a vertex
+  <C>[a, b]</C> has label <C>(a - 1) * |V2| + b</C> in the  output). There is
+  an edge from <C>[a, b]</C> to <C>[c, d]</C> precisely when the following
+  two conditions are satisfied:
+  <List>
+    <Item>
+      The vertices <C>a</C> and <C>c</C> of <A>D1</A> are equal if and only if
+      the vertices <C>b</C> and <C>d</C> of <A>D2</A> are equal.
+    </Item>
+    <Item>
+      There is an edge from <C>a</C> to <C>c</C> in <A>D1</A> if and only if
+      there is an edge from <C>b</C> to <C>d</C> in <A>D2</A>.
+    </Item>
+  </List>
+
+  Notably, the complete (with loops) subdigraphs of <C>MPD</C> are
+  precisely the partial isomorphisms from <A>D1</A> to <A>D2</A>.
+
+  <Example><![CDATA[
+gap> ModularProduct(Digraph([[1], [1, 2]]), Digraph([[], [2]]));
+<immutable digraph with 4 vertices, 4 edges>
+gap> OutNeighbours(last);
+[ [ 4 ], [ 2, 3 ], [  ], [ 4 ] ]
+gap> ModularProduct(PetersenGraph(), DigraphSymmetricClosure(CycleDigraph(5)));
+<immutable digraph with 50 vertices, 950 edges>
+gap> ModularProduct(NullDigraph(0), CompleteDigraph(10));
+<immutable empty digraph with 0 vertices>
+]]></Example>
+</Description>
+</ManSection>
+<#/GAPDoc>

gap/oper.gd

@@ -41,6 +41,8 @@ DeclareGlobalFunction("DigraphJoin");
 DeclareGlobalFunction("DigraphEdgeUnion");
 DeclareGlobalFunction("DigraphCartesianProduct");
 DeclareGlobalFunction("DigraphDirectProduct");
+DeclareOperation("ModularProduct", [IsDigraph, IsDigraph]);
+
 DeclareGlobalFunction("DIGRAPHS_CombinationOperProcessArgs");
 
 # 4. Actions . . .

gap/oper.gi

@@ -13,7 +13,7 @@
 #
 # 1.  Adding and removing vertices
 # 2.  Adding, removing, and reversing edges
-# 3.  Adding and removing vertices
+# 3.  Ways of combining digraphs
 # 4.  Actions
 # 5.  Substructures and quotients
 # 6.  In and out degrees, neighbours, and edges of vertices
@@ -612,6 +612,51 @@ function(arg)
   return D;
 end);
 
+InstallMethod(ModularProduct, "for a digraph and digraph",
+[IsDigraph, IsDigraph],
+function(D1, D2)
+  local m, n, E1, E2, edges, next, map, u, v, w, x;
+
+  if IsMultiDigraph(D1) or IsMultiDigraph(D2) then
+    ErrorNoReturn("ModularProduct does not support multidigraphs,");
+  fi;
+
+  m := DigraphNrVertices(D1);
+  n := DigraphNrVertices(D2);
+
+  E1 := DigraphDual(D1);
+  E2 := DigraphDual(D2);
+
+  edges := EmptyPlist(m * n);
+  next  := 0;
+
+  map := function(a, b)
+    return (a - 1) * n + b;
+  end;
+
+  for u in [1 .. m] do
+    for v in [1 .. n] do
+      next := next + 1;
+      edges[next] := [];
+      for w in OutNeighbours(D1)[u] do
+        for x in OutNeighbours(D2)[v] do
+          if (u = w) = (v = x) then
+            Add(edges[next], map(w, x));
+          fi;
+        od;
+      od;
+      for w in OutNeighbours(E1)[u] do
+        for x in OutNeighbours(E2)[v] do
+          if (u = w) = (v = x) then
+            Add(edges[next], map(w, x));
+          fi;
+        od;
+      od;
+    od;
+  od;
+  return DigraphNC(edges);
+end);
+
 ###############################################################################
 # 4. Actions
 ###############################################################################

tst/standard/oper.tst

@@ -2043,6 +2043,74 @@ gap> DigraphDijkstra(gr, 1, 2);
 gap> DigraphDijkstra(gr, 1, 3);
 [ [ 0, 1, 1, 1 ], [ -1, 1, 1, 1 ] ]
 
+#ModularProduct
+gap> ModularProduct(NullDigraph(0), CompleteDigraph(10));
+<immutable empty digraph with 0 vertices>
+gap> ModularProduct(PetersenGraph(), CompleteDigraph(10));
+<immutable digraph with 100 vertices, 2800 edges>
+gap> ModularProduct(NullDigraph(10), CompleteDigraph(10));
+<immutable digraph with 100 vertices, 100 edges>
+gap> ModularProduct(Digraph([[1], [1, 2]]), Digraph([[], [2]]));
+<immutable digraph with 4 vertices, 4 edges>
+gap> OutNeighbours(last);
+[ [ 4 ], [ 2, 3 ], [  ], [ 4 ] ]
+gap> ModularProduct(PetersenGraph(), DigraphSymmetricClosure(CycleDigraph(5)));
+<immutable digraph with 50 vertices, 950 edges>
+gap> OutNeighbours(last);
+[ [ 7, 10, 22, 25, 27, 30, 1, 13, 14, 18, 19, 33, 34, 38, 39, 43, 44, 48, 49 ]
+    , [ 6, 8, 21, 23, 26, 28, 2, 14, 15, 19, 20, 34, 35, 39, 40, 44, 45, 49, 
+      50 ], 
+  [ 7, 9, 22, 24, 27, 29, 3, 11, 15, 16, 20, 31, 35, 36, 40, 41, 45, 46, 50 ],
+  [ 8, 10, 23, 25, 28, 30, 4, 11, 12, 16, 17, 31, 32, 36, 37, 41, 42, 46, 47 ]
+    , [ 6, 9, 21, 24, 26, 29, 5, 12, 13, 17, 18, 32, 33, 37, 38, 42, 43, 47, 
+      48 ], 
+  [ 2, 5, 12, 15, 32, 35, 6, 18, 19, 23, 24, 28, 29, 38, 39, 43, 44, 48, 49 ],
+  [ 1, 3, 11, 13, 31, 33, 7, 19, 20, 24, 25, 29, 30, 39, 40, 44, 45, 49, 50 ],
+  [ 2, 4, 12, 14, 32, 34, 8, 16, 20, 21, 25, 26, 30, 36, 40, 41, 45, 46, 50 ],
+  [ 3, 5, 13, 15, 33, 35, 9, 16, 17, 21, 22, 26, 27, 36, 37, 41, 42, 46, 47 ],
+  [ 1, 4, 11, 14, 31, 34, 10, 17, 18, 22, 23, 27, 28, 37, 38, 42, 43, 47, 48 ]
+    , [ 7, 10, 17, 20, 37, 40, 3, 4, 11, 23, 24, 28, 29, 33, 34, 43, 44, 48, 
+      49 ], [ 6, 8, 16, 18, 36, 38, 4, 5, 12, 24, 25, 29, 30, 34, 35, 44, 45, 
+      49, 50 ], 
+  [ 7, 9, 17, 19, 37, 39, 1, 5, 13, 21, 25, 26, 30, 31, 35, 41, 45, 46, 50 ], 
+  [ 8, 10, 18, 20, 38, 40, 1, 2, 14, 21, 22, 26, 27, 31, 32, 41, 42, 46, 47 ],
+  [ 6, 9, 16, 19, 36, 39, 2, 3, 15, 22, 23, 27, 28, 32, 33, 42, 43, 47, 48 ], 
+  [ 12, 15, 22, 25, 42, 45, 3, 4, 8, 9, 16, 28, 29, 33, 34, 38, 39, 48, 49 ], 
+  [ 11, 13, 21, 23, 41, 43, 4, 5, 9, 10, 17, 29, 30, 34, 35, 39, 40, 49, 50 ],
+  [ 12, 14, 22, 24, 42, 44, 1, 5, 6, 10, 18, 26, 30, 31, 35, 36, 40, 46, 50 ],
+  [ 13, 15, 23, 25, 43, 45, 1, 2, 6, 7, 19, 26, 27, 31, 32, 36, 37, 46, 47 ], 
+  [ 11, 14, 21, 24, 41, 44, 2, 3, 7, 8, 20, 27, 28, 32, 33, 37, 38, 47, 48 ], 
+  [ 2, 5, 17, 20, 47, 50, 8, 9, 13, 14, 21, 28, 29, 33, 34, 38, 39, 43, 44 ], 
+  [ 1, 3, 16, 18, 46, 48, 9, 10, 14, 15, 22, 29, 30, 34, 35, 39, 40, 44, 45 ],
+  [ 2, 4, 17, 19, 47, 49, 6, 10, 11, 15, 23, 26, 30, 31, 35, 36, 40, 41, 45 ],
+  [ 3, 5, 18, 20, 48, 50, 6, 7, 11, 12, 24, 26, 27, 31, 32, 36, 37, 41, 42 ], 
+  [ 1, 4, 16, 19, 46, 49, 7, 8, 12, 13, 25, 27, 28, 32, 33, 37, 38, 42, 43 ], 
+  [ 2, 5, 37, 40, 42, 45, 8, 9, 13, 14, 18, 19, 23, 24, 26, 33, 34, 48, 49 ], 
+  [ 1, 3, 36, 38, 41, 43, 9, 10, 14, 15, 19, 20, 24, 25, 27, 34, 35, 49, 50 ],
+  [ 2, 4, 37, 39, 42, 44, 6, 10, 11, 15, 16, 20, 21, 25, 28, 31, 35, 46, 50 ],
+  [ 3, 5, 38, 40, 43, 45, 6, 7, 11, 12, 16, 17, 21, 22, 29, 31, 32, 46, 47 ], 
+  [ 1, 4, 36, 39, 41, 44, 7, 8, 12, 13, 17, 18, 22, 23, 30, 32, 33, 47, 48 ], 
+  [ 7, 10, 42, 45, 47, 50, 3, 4, 13, 14, 18, 19, 23, 24, 28, 29, 31, 38, 39 ],
+  [ 6, 8, 41, 43, 46, 48, 4, 5, 14, 15, 19, 20, 24, 25, 29, 30, 32, 39, 40 ], 
+  [ 7, 9, 42, 44, 47, 49, 1, 5, 11, 15, 16, 20, 21, 25, 26, 30, 33, 36, 40 ], 
+  [ 8, 10, 43, 45, 48, 50, 1, 2, 11, 12, 16, 17, 21, 22, 26, 27, 34, 36, 37 ],
+  [ 6, 9, 41, 44, 46, 49, 2, 3, 12, 13, 17, 18, 22, 23, 27, 28, 35, 37, 38 ], 
+  [ 12, 15, 27, 30, 47, 50, 3, 4, 8, 9, 18, 19, 23, 24, 33, 34, 36, 43, 44 ], 
+  [ 11, 13, 26, 28, 46, 48, 4, 5, 9, 10, 19, 20, 24, 25, 34, 35, 37, 44, 45 ],
+  [ 12, 14, 27, 29, 47, 49, 1, 5, 6, 10, 16, 20, 21, 25, 31, 35, 38, 41, 45 ],
+  [ 13, 15, 28, 30, 48, 50, 1, 2, 6, 7, 16, 17, 21, 22, 31, 32, 39, 41, 42 ], 
+  [ 11, 14, 26, 29, 46, 49, 2, 3, 7, 8, 17, 18, 22, 23, 32, 33, 40, 42, 43 ], 
+  [ 17, 20, 27, 30, 32, 35, 3, 4, 8, 9, 13, 14, 23, 24, 38, 39, 41, 48, 49 ], 
+  [ 16, 18, 26, 28, 31, 33, 4, 5, 9, 10, 14, 15, 24, 25, 39, 40, 42, 49, 50 ],
+  [ 17, 19, 27, 29, 32, 34, 1, 5, 6, 10, 11, 15, 21, 25, 36, 40, 43, 46, 50 ],
+  [ 18, 20, 28, 30, 33, 35, 1, 2, 6, 7, 11, 12, 21, 22, 36, 37, 44, 46, 47 ], 
+  [ 16, 19, 26, 29, 31, 34, 2, 3, 7, 8, 12, 13, 22, 23, 37, 38, 45, 47, 48 ], 
+  [ 22, 25, 32, 35, 37, 40, 3, 4, 8, 9, 13, 14, 18, 19, 28, 29, 43, 44, 46 ], 
+  [ 21, 23, 31, 33, 36, 38, 4, 5, 9, 10, 14, 15, 19, 20, 29, 30, 44, 45, 47 ],
+  [ 22, 24, 32, 34, 37, 39, 1, 5, 6, 10, 11, 15, 16, 20, 26, 30, 41, 45, 48 ],
+  [ 23, 25, 33, 35, 38, 40, 1, 2, 6, 7, 11, 12, 16, 17, 26, 27, 41, 42, 49 ], 
+  [ 21, 24, 31, 34, 36, 39, 2, 3, 7, 8, 12, 13, 17, 18, 27, 28, 42, 43, 50 ] ]
+
 #DIGRAPHS_UnbindVariables
 gap> Unbind(a);
 gap> Unbind(adj);