Quelle attr.gi Sprache: unbekannt

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##
##  attr.gi
##  Copyright (C) 2014-21                                James D. Mitchell
##
##  Licensing information can be found in the README file of this package.
##
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##

InstallMethod(DigraphNrVertices, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep], DIGRAPH_NR_VERTICES);

InstallGlobalFunction(OutNeighbors, OutNeighbours);

# The next method is (yet another) DFS which simultaneously computes:
# 1. *articulation points* as described in
#   https://www.eecs.wsu.edu/~holder/courses/CptS223/spr08/slides/graphapps.pdf
# 2. *bridges* as described in https://stackoverflow.com/q/28917290/
#   (this is a minor adaption of the algorithm described in point 1).
# 3. a *strong orientation* as alluded to somewhere on the internet that I can
#    no longer find. It's essentially just "orient every edge in the DFS tree
#    away from the root, and every other edge (back edges) from the node with
#    higher `pre` value to the one with lower `pre` value (i.e. they point
#    backwards from later nodes in the DFS to earlier ones). If the graph is
#    bridgeless, then it is guaranteed that the orientation of the last
#    sentence is strongly connected."

BindGlobal("DIGRAPHS_ArticulationPointsBridgesStrongOrientation",
function(D)
  local N, copy, articulation_points, bridges, orientation, nbs, counter, pre,
        low, nr_children, stack, u, v, i, w, connected;

  N := DigraphNrVertices(D);

  if HasIsConnectedDigraph(D) and not IsConnectedDigraph(D) then
    # not connected, no articulation points, no bridges, no strong orientation
    return [false, [], [], fail];
  elif N < 2 then
    # connected, no articulation points (removing 0 or 1 nodes does not make
    # the graph disconnected), no bridges, strong orientation (since
    # the digraph with 0 nodes is strongly connected).
    return [true, [], [], D];
  elif not IsSymmetricDigraph(D) then
    copy := DigraphSymmetricClosure(DigraphMutableCopyIfMutable(D));
    MakeImmutable(copy);
  else
    copy := D;
  fi;

  # outputs
  articulation_points := [];
  bridges             := [];
  orientation         := List([1 .. N], x -> BlistList([1 .. N], []));

  # Get out-neighbours once, to avoid repeated copying for mutable digraphs.
  nbs := OutNeighbours(copy);

  # number of nodes encountered in the search so far
  counter := 0;

  # the order in which the nodes are visited, -1 indicates "not yet visited".
  pre := ListWithIdenticalEntries(N, -1);

  # low[i] is the lowest value in pre currently reachable from node i.
  low := [];

  # nr_children of node 1, for articulation points the root node (1) is an
  # articulation point if and only if it has at least 2 children.
  nr_children := 0;

  stack := Stack();
  u := 1;
  v := 1;
  i := 0;

  repeat
    if pre[v] <> -1 then
      # backtracking
      i := Pop(stack);
      v := Pop(stack);
      u := Pop(stack);
      w := nbs[v][i];

      if v <> 1 and low[w] >= pre[v] then
        AddSet(articulation_points, v);
      fi;
      if low[w] = pre[w] then
        Add(bridges, [v, w]);
      fi;
      if low[w] < low[v] then
        low[v] := low[w];
      fi;
    else
      # diving - part 1
      counter := counter + 1;
      pre[v]  := counter;
      low[v]  := counter;
    fi;
    i := PositionProperty(nbs[v], w -> w <> v, i);
    while i <> fail do
      w := nbs[v][i];
      if pre[w] <> -1 then
        # v -> w is a back edge
        if w <> u and pre[w] < low[v] then
          low[v] := pre[w];
        fi;
        orientation[v][w] := not orientation[w][v];
        i := PositionProperty(nbs[v], w -> w <> v, i);
      else
        # diving - part 2
        if v = 1 then
          nr_children := nr_children + 1;
        fi;
        orientation[v][w] := true;
        Push(stack, u);
        Push(stack, v);
        Push(stack, i);
        u := v;
        v := w;
        i := 0;
        break;
      fi;
    od;
  until Size(stack) = 0;

  if counter = DigraphNrVertices(D) then
    connected := true;
    if nr_children > 1 then
      # The `AddSet` is not really needed, and could just be `Add`, since 1
      # should not already be in articulation_points. But let's use AddSet
      # to keep the output sorted.
      AddSet(articulation_points, 1);
    fi;
    if not IsEmpty(bridges) then
      orientation := fail;
    else
      orientation := DigraphByAdjacencyMatrix(DigraphMutabilityFilter(D),
                                              orientation);
    fi;
  else
    connected           := false;
    articulation_points := [];
    bridges             := [];
    orientation         := fail;
  fi;
  if IsImmutableDigraph(D) then
    SetIsConnectedDigraph(D, connected);
    SetArticulationPoints(D, articulation_points);
    SetBridges(D, bridges);
    if IsSymmetricDigraph(D) then
      SetStrongOrientationAttr(D, orientation);
    fi;
  fi;
  return [connected, articulation_points, bridges, orientation];
end);

InstallMethod(ArticulationPoints, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
D -> DIGRAPHS_ArticulationPointsBridgesStrongOrientation(D)[2]);

InstallMethod(Bridges, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
D -> DIGRAPHS_ArticulationPointsBridgesStrongOrientation(D)[3]);

InstallMethodThatReturnsDigraph(StrongOrientation,
"for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  if not IsSymmetricDigraph(D) then
    ErrorNoReturn("not yet implemented");
  fi;
  return DIGRAPHS_ArticulationPointsBridgesStrongOrientation(D)[4];
end);

# Utility function to calculate the maximal independent sets (as BLists) of a
# subgraph induced by removing a set of vertices.
BindGlobal("DIGRAPHS_MaximalIndependentSetsSubtractedSet",
function(I, subtracted_set, size_bound)
  local induced_mis, temp, i;
  # First remove all vertices in the subtracted set from each MIS
  temp := List(I, i -> DifferenceBlist(i, subtracted_set));
  induced_mis := [];
  # Then remove any sets which are no longer maximal
  # Sort in decreasing size
  Sort(temp, {x, y} -> SizeBlist(x) > SizeBlist(y));
  # Then check elements from back to front for if they are a subset
  for i in temp do
    if SizeBlist(i) <= size_bound and
        ForAll(induced_mis, x -> not IsSubsetBlist(x, i)) then
      Add(induced_mis, i);
    fi;
  od;
  return induced_mis;
end
);

InstallMethod(DIGRAPHS_AbsorbingMarkovChain,
"for a digraph",
[IsDigraph],
function(D)
  # Helper for DigraphAbsorptionProbabilities and DigraphAbsorptionExpectedSteps
  local scc, is_sink_comp, transient_vertices, i, comp, v, sink_comps,
        nr_transients, transient_mat, absorption_mat, neighbours, chance, w,
        w_comp, sink_comp_index, j, fundamental_mat;
  scc := DigraphStronglyConnectedComponents(D);

  # Find the "sink components" (components from which there is no escape)
  # We could use QuotientDigraph and DigraphSinks here, but this avoids copying.
  is_sink_comp := ListWithIdenticalEntries(Length(scc.comps), true);
  transient_vertices := [];
  for i in [1 .. Length(scc.comps)] do
    comp := scc.comps[i];
    for v in comp do
      if ForAny(OutNeighboursOfVertex(D, v), w -> not w in comp) then
        is_sink_comp[i] := false;
        Append(transient_vertices, comp);
        break;
      fi;
    od;
  od;
  sink_comps := Positions(is_sink_comp, true);
  nr_transients := Length(transient_vertices);

  # If no transient vertices, then we can't return anything interesting.
  if nr_transients = 0 then
    return rec(transient_vertices := transient_vertices,
               fundamental_mat := [],
               sink_comps := sink_comps,
               absorption_mat := []);
  fi;

  # transient_mat[i][j] is the chance of going
  # from transient vertex i to transient vertex j
  transient_mat := NullMat(nr_transients, nr_transients);

  # absorption_mat[i][j] is the chance of going
  # from transient vertex i to sink component j
  absorption_mat := NullMat(nr_transients, Length(sink_comps));

  for i in [1 .. Length(transient_vertices)] do
    v := transient_vertices[i];
    neighbours := OutNeighboursOfVertex(D, v);
    chance := 1 / Length(neighbours);  # chance of following each edge
    for w in neighbours do
      w_comp := scc.id[w];
      if is_sink_comp[w_comp] then
        # w is in a sink component
        sink_comp_index := Position(sink_comps, w_comp);
        Assert(1, w in scc.comps[sink_comps[sink_comp_index]]);
        absorption_mat[i][sink_comp_index] :=
            absorption_mat[i][sink_comp_index] + chance;
      else
        # w is a transient vertex
        j := Position(transient_vertices, w);
        transient_mat[i][j] := transient_mat[i][j] + chance;
      fi;
    od;
  od;

  # "Fundamental matrix": mat[i][j] is the expected number of visits to each
  # transient vertex j given a random walk starting at transient vertex i.
  # Compute this using formula (I - Q)^-1
  fundamental_mat := Inverse(IdentityMat(nr_transients) - transient_mat);

  # Return all info needed for further computation in DigraphAbsorption* methods
  return rec(transient_vertices := transient_vertices,
             fundamental_mat := fundamental_mat,
             sink_comps := sink_comps,
             absorption_mat := absorption_mat);
end);

InstallMethod(DigraphAbsorptionProbabilities,
"for a digraph",
[IsDigraph],
function(D)
  local scc, markov, chances_of_absorption, output, sink_comps, comp_no, v,
        transient_vertices, i, j, c;
  # Strongly connected components are an important part of definition
  scc := DigraphStronglyConnectedComponents(D);

  # One SCC only (or none): all vertices stay in it with probability 1.
  if Length(scc.comps) <= 1 then
    return ListWithIdenticalEntries(DigraphNrVertices(D), [1]);
  fi;

  # Get data about absorbing Markov chain represented by this digraph
  markov := DIGRAPHS_AbsorbingMarkovChain(D);

  # Calculate chances of absorption
  # Rows are transient vertices, columns are absorbing SCCs
  if Length(markov.transient_vertices) = 0 then
    chances_of_absorption := [];
  else
    chances_of_absorption := markov.fundamental_mat * markov.absorption_mat;
  fi;

  # Convert to output format
  # Rows are all vertices, columns are all SCCs
  output := NullMat(DigraphNrVertices(D), Length(scc.comps));

  # Non-transient vertices stay in their own SCC with probability 1
  sink_comps := markov.sink_comps;
  for comp_no in sink_comps do
    for v in scc.comps[comp_no] do
      output[v][comp_no] := 1;
    od;
  od;

  # Transient vertices have chances as calculated in Markov code
  transient_vertices := markov.transient_vertices;
  for i in [1 .. Length(transient_vertices)] do
    v := transient_vertices[i];
    for j in [1 .. Length(sink_comps)] do
      c := sink_comps[j];
      output[v][c] := chances_of_absorption[i][j];
    od;
  od;

  return output;
end);

InstallMethod(DigraphAbsorptionExpectedSteps,
"for a digraph",
[IsDigraph],
function(D)
  local markov, N, transient_expected_steps, out, i;
  if DigraphNrVertices(D) = 0 then
    return [];
  fi;
  markov := DIGRAPHS_AbsorbingMarkovChain(D);
  N := markov.fundamental_mat;
  if Length(N) = 0 then  # no transient vertices
    transient_expected_steps := [];
  else
    transient_expected_steps := N * ListWithIdenticalEntries(Length(N), 1);
  fi;
  out := ListWithIdenticalEntries(DigraphNrVertices(D), 0);
  for i in [1 .. Length(transient_expected_steps)] do
    out[markov.transient_vertices[i]] := transient_expected_steps[i];
  od;
  return out;
end);

BindGlobal("DIGRAPHS_ChromaticNumberLawler",
function(D)
  local n, vertices, subset_colours, s, S, i, I, subset_iter, x,
  mis, subset_mis;
  n := DigraphNrVertices(D);
  vertices := List(DigraphVertices(D));
  # Store all the Maximal Independent Sets, which can later be used for
  # calculating the maximal independent sets of induced subgraphs.
  mis := DigraphMaximalIndependentSets(D);
  # Convert each MIS to a Blist
  mis := List(mis, x -> BlistList(vertices, x));
  # Store current best colouring for each subset
  subset_colours := ListWithIdenticalEntries(2 ^ n, infinity);
  # Empty set can be colouring with only one colour.
  subset_colours[1] := 0;
  # Iterator for blist subsets.
  subset_iter := ListWithIdenticalEntries(n, [false, true]);
  subset_iter := IteratorOfCartesianProduct2(subset_iter);
  # Skip the first one, which should be the empty set.
  S := NextIterator(subset_iter);
  # Iterate over all vertex subsets.
  for S in subset_iter do
    # Cartesian iterator is ascending lexicographically, but we want reverse
    # lexicographic ordering. We treat this as iterating over the complement.
    # Index the current subset that is being iterated over.
    s := 1;
    for x in [1 .. n] do
      # Need to negate, as we are iterating over the complement.
      if not S[x] then
        s := s + 2 ^ (x - 1);
      fi;
    od;
    # Iterate over the maximal independent sets of V[S]
    subset_mis :=
      DIGRAPHS_MaximalIndependentSetsSubtractedSet(mis, S, infinity);
    # Flip the list, as we now need the actual set.
    FlipBlist(S);
    for I in subset_mis do
        # Calculate S \ I. This is destructive, but is undone.
        SubtractBlist(S, I);
        # Index S \ I
        i := 1;
        for x in [1 .. n] do
          if S[x] then
            i := i + 2 ^ (x - 1);
          fi;
        od;
        # The chromatic number of this subset is the minimum value of all
        # the maximal independent subsets of D[S].
        subset_colours[s] := Minimum(subset_colours[s], subset_colours[i] + 1);
        # Undo the changes to the subset.
        UniteBlist(S, I);
    od;
  od;
  return subset_colours[2 ^ n];
end
);

BindGlobal("DIGRAPHS_UnderThreeColourable",
function(D)
  local nr;
  nr := DigraphNrVertices(D);
  if DigraphHasLoops(D) then
    ErrorNoReturn("the argument <D> must be a digraph with no loops,");
  elif nr = 0 then
    return 0;  # chromatic number = 0 iff <D> has 0 verts
  elif IsNullDigraph(D) then
    return 1;  # chromatic number = 1 iff <D> has >= 1 verts & no edges
  elif IsBipartiteDigraph(D) then
    return 2;  # chromatic number = 2 iff <D> has >= 2 verts & is bipartite
               # <D> has at least 2 vertices at this stage
  elif DigraphColouring(D, 3) <> fail then
               # Check if there is a 3 colouring
    return 3;
  fi;
  return infinity;
end
);

BindGlobal("DIGRAPHS_ChromaticNumberByskov",
function(D)
  local n, a, vertices, subset_colours, S, i, j, I, subset_iter,
  index_subsets, vertex_blist, k, MIS;

  n := DigraphNrVertices(D);
  vertices := DigraphVertices(D);
  vertex_blist := BlistList(vertices, vertices);
  # Store all the Maximal Independent Sets, which can later be used for
  # calculating the maximal independent sets of induced subgraphs.
  MIS := DigraphMaximalIndependentSets(D);
  # Convert each MIS to a Blist
  MIS := List(MIS, x -> BlistList(vertices, x));
  # Store current best colouring for each subset
  subset_colours := ListWithIdenticalEntries(2 ^ n, infinity);
  # Empty set is 0 colourable
  subset_colours[1] := 0;
  # Function to index the subsets of the vertices of D
  index_subsets := function(subset)
    local x, index;
    index := 1;
    for x in [1 .. n] do
      if subset[x] then
        index := index + 2 ^ (x - 1);
      fi;
    od;
    return index;
  end;
  # Iterate over vertex subsets
  subset_iter := IteratorOfCombinations(vertices);
  # Skip the first one, which should be the empty set
  S := NextIterator(subset_iter);
  Assert(1, IsEmpty(S), "First set from iterator should be the empty set");
  # First find the 3 colourable subgraphs of D
  for S in subset_iter do
    a := DIGRAPHS_UnderThreeColourable(InducedSubdigraph(D, S));
    S := BlistList(vertices, S);
    i := index_subsets(S);
    # Mark this as three or less colourable if it is.
    subset_colours[i] := Minimum(a, subset_colours[i]);
  od;
  # Process 4 colourable subgraphs
  for I in MIS do
    SubtractBlist(vertex_blist, I);
    # Iterate over all subsets of V(D) \ I as blists
    # This is done by taking the cartesian product of n copies of [true, false]
    # or [true] if the vertex is in I. The [true] is used as each element will
    # be flipped to get reverse lexicographic ordering.
    subset_iter := EmptyPlist(n);
    for i in [1 .. n] do
      if I[i] then
        subset_iter[i] := [true];
      else
        subset_iter[i] := [true, false];
      fi;
    od;
    subset_iter := IteratorOfCartesianProduct2(subset_iter);
    # Skip the empty set.
    NextIterator(subset_iter);
    for S in subset_iter do
      FlipBlist(S);
      i := index_subsets(S);
      if subset_colours[i] = 3 then
        # Index union of S and I
        j := index_subsets(UnionBlist(S, I));
        subset_colours[j] := Minimum(subset_colours[j], 4);
      fi;
    od;
    # Undo the changes made.
    UniteBlist(vertex_blist, I);
  od;
  # Iterate over vertex subset blists.
  subset_iter := ListWithIdenticalEntries(n, [true, false]);
  subset_iter := IteratorOfCartesianProduct2(subset_iter);
  # Skip the first one, which should be the empty set
  S := NextIterator(subset_iter);
  for S in subset_iter do
    # Cartesian iteratator goes in lexicographic order, but we want reverse.
    FlipBlist(S);
    # Index the current subset that is being iterated over
    i := index_subsets(S);
    if 4 <= subset_colours[i] and subset_colours[i] < infinity then
      k := SizeBlist(S) / subset_colours[i];
      # Iterate over the maximal independent sets of D[V \ S]
      for I in DIGRAPHS_MaximalIndependentSetsSubtractedSet(MIS, S, k) do
        # Index S union I
        j := index_subsets(UnionBlist(S, I));
        subset_colours[j] := Minimum(subset_colours[j], subset_colours[i] + 1);
      od;
    fi;
  od;
  return subset_colours[2 ^ n];
end
);

BindGlobal("DIGRAPHS_ChromaticNumberZykov",
function(D)
  local nr, ZykovReduce, chrom;
  nr := DigraphNrVertices(D);
  # Recursive function call
  ZykovReduce := function(D)
    local nr, D_contract, vertices, v, x, y, i, j, adjacent;
    nr := DigraphNrVertices(D);
    # Update upper bound if possible.
    chrom := Minimum(nr, chrom);
    # Leaf nodes are either complete graphs or cliques that have size equal to
    # the current upper bound. The chromatic number is then the smallest clique
    # found.
    # Cliques finder arguments:
    # digraph = D - The graph
    # hook = fail - hook is not required
    # user_param = [] - user_param is a list as hook is fail
    # limit = 1 - We only need one clique
    # include = exclude = [] - We check all vertices
    # max = false - This clique need not be maximal
    # size = chrom - We want a clique the size of our upper bound
    # reps = true - As we only care about the existence of the clique,
    # we can instead search for representatives which is more efficient.
    if not IsCompleteDigraph(D) and IsEmpty(CliquesFinder(D, fail, [], 1, [],
                                                          [], false, chrom,
                                                          true)) then
      # Sort vertices by degree, so that higher degree vertices are picked first
      # Picking higher degree vertices will make it more likely a clique will
      # form in one of the modified graphs, which will terminate the recursion.
      vertices := DigraphWelshPowellOrder(D);
      # Get the adjacency function
      adjacent := DigraphAdjacencyFunction(D);
      # Choose two non-adjacent vertices x, y
      for i in [1 .. nr] do
        x := vertices[i];
        # Search for the first vertex not adjacent to all others
        # This is guaranteed to exist as D is not the complete graph.
        if OutDegreeOfVertex(D, x) < nr - 1 then
          # Now search for a non-adjacent vertex, prioritising higher degree
          # ones
          for j in [i + 1 .. nr] do
            y := vertices[j];
            if not adjacent(x, y) then
              break;
            fi;
          od;
          break;
        fi;
      od;
      Assert(1, x <> y, "x and y must be different");
      # Colour the vertex contraction.
      # A contraction of a graph effectively merges two non adjacent vertices
      # into a single new vertex with the edges merged.
      # We merge y into x, keeping x.

      # We could potentially use quotient digraph here, but the increased
      # generality might cause this to get slower.
      D_contract := DigraphMutableCopy(D);
      for v in vertices do
         # Iterate over all vertices that are not x or y
         if v = x or v = y then
           continue;
         fi;
         # Add any edge that involves y, but not already x to avoid duplication.
         if adjacent(v, y) and not adjacent(v, x) then
            DigraphAddEdge(D_contract, x, v);
            DigraphAddEdge(D_contract, v, x);
         fi;
      od;
      DigraphRemoveVertex(D_contract, y);
      ZykovReduce(D_contract);
      # Colour the edge addition
      # This just adds symmetric edges between x and y;
      DigraphAddEdge(D, [x, y]);
      DigraphAddEdge(D, [y, x]);
      ZykovReduce(D);
      # Undo changes to the graph
      DigraphRemoveEdge(D, [x, y]);
      DigraphRemoveEdge(D, [y, x]);
    fi;
  end;
  # Algorithm requires an undirected graph without multiple edges.
  D := DigraphMutableCopy(D);
  D := DigraphRemoveAllMultipleEdges(D);
  D := DigraphSymmetricClosure(D);
  # Use greedy colouring as an upper bound
  chrom := RankOfTransformation(DigraphGreedyColouring(D), nr);
  ZykovReduce(D);
  return chrom;
end
);

BindGlobal("DIGRAPHS_ChromaticNumberChristofides",
function(D)
  local nr, I, n, T, b, unprocessed, i, v_without_t, j, u, min_occurrences,
  cur_occurrences, chrom, colouring, stack, vertices;

  nr := DigraphNrVertices(D);
  vertices := List(DigraphVertices(D));
  # Initialise the required variables.
  # Calculate all maximal independent sets of D.
  I := DigraphMaximalIndependentSets(D);
  # Convert each MIS into a BList
  I := List(I, i -> BlistList(vertices, i));
  # Upper bound for chromatic number.
  chrom := nr;
  # Set of vertices of D not in the current subgraph at level n.
  T := ListWithIdenticalEntries(nr, false);
  # Current search level of the subgraph tree.
  n := 0;
  # The maximal independent sets of V \ T at level n.
  b := [ListWithIdenticalEntries(nr, false)];
  # Number of unprocessed MIS's of V \ T from level 1 to n
  unprocessed := ListWithIdenticalEntries(nr, 0);
  # Would be jth colour class of the chromatic colouring of G.
  colouring := List([1 .. nr], i -> BlistList(vertices, [i]));
  # Stores current unprocessed MIS's of V \ T at level 1 to level n
  stack := [];
  # Now perform the search.
  repeat
    # Step 2
    if n < chrom then
      # Step 3
      # If V = T then we've reached a null subgraph
      if SizeBlist(T) = nr then
        chrom := n;
        SubtractBlist(T, b[n + 1]);
        for i in [1 .. chrom] do
          colouring[i] := b[i];
          # TODO set colouring attribute
        od;
      else
        # Step 4
        # Compute the maximal independent sets of V \ T
        v_without_t := DIGRAPHS_MaximalIndependentSetsSubtractedSet(I, T,
                                                                    infinity);
        # Step 5
        # Pick u in V \ T such that u is in the fewest maximal independent sets.
        u := -1;
        min_occurrences := infinity;
        # Flip T to get V \ T
        FlipBlist(T);
        # Convert to list to iterate over the vertices.
        for i in ListBlist(vertices, T) do
          # Count how many times this vertex appears in a MIS
          cur_occurrences := Number(v_without_t, j -> j[i]);
          if cur_occurrences < min_occurrences then
            min_occurrences := cur_occurrences;
            u := i;
          fi;
        od;
        # Revert changes to T
        FlipBlist(T);
        Assert(1, u <> -1, "Vertex must be picked");
        # Remove maximal independent sets not containing u.
        v_without_t := Filtered(v_without_t, x -> x[u]);
        # Add these MISs to the stack
        Append(stack, v_without_t);
        # Search has moved one level deeper
        n := n + 1;
        unprocessed[n] := Length(v_without_t);
      fi;
    else
      # if n >= g then T = T \ b[n]
      # This exceeds the current best bound, so stop search.
      SubtractBlist(T, b[n + 1]);
    fi;
    # Step 6
    while n <> 0 do
      # step 7
      if unprocessed[n] = 0 then
        n := n - 1;
        SubtractBlist(T, b[n + 1]);
      else
        # Step 8
        # take an element from the top of the stack
        i := Remove(stack);
        unprocessed[n] := unprocessed[n] - 1;
        b[n + 1] := i;
        UniteBlist(T, i);
        break;
      fi;
    od;
  until n = 0;
  return chrom;
end
);

InstallMethod(ChromaticNumber, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local nr, comps, upper, chrom, tmp_comps, tmp_upper, n, comp, bound, clique,
  c, i, greedy_bound, brooks_bound;
  nr := DigraphNrVertices(D);

  if DigraphHasLoops(D) then
    ErrorNoReturn("the argument <D> must be a digraph with no loops,");
  elif nr = 0 then
    return 0;  # chromatic number = 0 iff <D> has 0 verts
  elif IsNullDigraph(D) then
    return 1;  # chromatic number = 1 iff <D> has >= 1 verts & no edges
  elif IsBipartiteDigraph(D) then
    return 2;  # chromatic number = 2 iff <D> has >= 2 verts & is bipartite
               # <D> has at least 2 vertices at this stage
  fi;

  # Value option for alternative dispatch
  if ValueOption("lawler") <> fail then
    return DIGRAPHS_ChromaticNumberLawler(D);
  elif ValueOption("byskov") <> fail then
    return DIGRAPHS_ChromaticNumberByskov(D);
  elif ValueOption("zykov") <> fail then
    return DIGRAPHS_ChromaticNumberZykov(D);
  elif ValueOption("christofides") <> fail then
    return DIGRAPHS_ChromaticNumberChristofides(D);
  fi;

  # The chromatic number of <D> is at least 3 and at most nr
  D := DigraphMutableCopy(D);
  D := DigraphRemoveAllMultipleEdges(D);
  D := DigraphSymmetricClosure(D);
  MakeImmutable(D);

  if IsCompleteDigraph(D) then
    # chromatic number = nr iff <D> has >= 2 verts & this cond.
    return nr;
  elif nr = 4 then
    # if nr = 4, then 3 is only remaining possible chromatic number
    return 3;
  elif 2 * nr = DigraphNrEdges(D)
      and IsRegularDigraph(D) and Length(OutNeighboursOfVertex(D, 1)) = 2 then
    # <D> is an odd-length cycle graph
    return 3;
  fi;

  # The chromatic number of <D> is at least 3 and at most nr - 1

  # The variable <chrom> is the current best known lower bound for the
  # chromatic number of <D>.
  chrom := 3;

  # Prepare a list of connected components of D whose chromatic number we
  # do not yet know.
  if IsConnectedDigraph(D) then
    comps := [D];
    greedy_bound := RankOfTransformation(DigraphGreedyColouring(D), nr);
    brooks_bound := Maximum(OutDegrees(D));  # Brooks' theorem
    upper := [Minimum(greedy_bound, brooks_bound)];
    chrom := Maximum(CliqueNumber(D), chrom);
  else
    tmp_comps := [];
    tmp_upper := [];
    for comp in DigraphConnectedComponents(D).comps do
      n := Length(comp);
      if chrom < n then
        # If chrom >= n, then we can colour the vertices of comp using any n of
        # the required (at least) chrom colours, and we do not have to consider
        # comp.

        # Note that n > chrom >= 3 and so comp is not null, so no need to check
        # for that.
        comp := InducedSubdigraph(DigraphMutableCopy(D), comp);
        if IsCompleteDigraph(comp) then
          # Since n > chrom, this is an improved lower bound for the overall
          # chromatic number.
          chrom := n;
        elif not IsBipartiteDigraph(comp) then
          # If comp is bipartite, then its chromatic number is 2, and, since
          # the chromatic number of D is >= 3, this component can be
          # ignored.
          greedy_bound := RankOfTransformation(DigraphGreedyColouring(comp),
                                               DigraphNrVertices(comp));
          # Don't need to take odd cycles into account for Brooks' theorem,
          # since they are 3-colourable and the chromatic number of D is >= 3.
          brooks_bound := Maximum(OutDegrees(comp));
          bound := Minimum(greedy_bound, brooks_bound);
          if bound > chrom then
            # If bound <= chrom, then comp can be coloured by at most chrom
            # colours, and so we can ignore comp.
            clique := CliqueNumber(comp);
            if clique = bound then
              # The chromatic number of this component is known, and it can be
              # ignored, and clique = bound > chrom, and so clique is an
              # improved lower bound for the chromatic number of D.
              chrom := clique;
            else
              Add(tmp_comps, comp);
              Add(tmp_upper, bound);
              if clique > chrom then
                chrom := clique;
              fi;
            fi;
          fi;
        fi;
      fi;
    od;

    # Remove the irrelevant components since we have a possibly improved value
    # of chrom.
    comps := [];
    upper := [];

    for i in [1 .. Length(tmp_comps)] do
      if chrom < DigraphNrVertices(tmp_comps[i]) and chrom < tmp_upper[i] then
        Add(comps, tmp_comps[i]);
        Add(upper, tmp_upper[i]);
      fi;
    od;

    # Sort by size, since smaller components are easier to colour
    SortParallel(comps, upper, {x, y} -> Size(x) < Size(y));
  fi;
  for i in [1 .. Length(comps)] do
    # <c> is the current best upper bound for the chromatic number of comps[i]
    c := upper[i];
    while c > chrom and DigraphColouring(comps[i], c - 1) <> fail do
      c := c - 1;
    od;
    if c > chrom then
      chrom := c;
    fi;
  od;
  return chrom;
end);

#
# The following method is currently useless, as the OutNeighbours are computed
# and set whenever a digraph is created.  It could be reinstated later if we
# decide to allow digraphs to exist without known OutNeighbours.
#

# InstallMethod(OutNeighbours,
# "for a digraph with representative out neighbours and group",
# [IsDigraph and HasRepresentativeOutNeighbours and HasDigraphGroup],
# function(D)
#   local gens, sch, reps, out, trace, word, i, w;
#
#   gens := GeneratorsOfGroup(DigraphGroup(D));
#   sch  := DigraphSchreierVector(D);
#   reps := RepresentativeOutNeighbours(D);
#
#   out  := EmptyPlist(DigraphNrVertices(D));
#
#   for i in [1 .. Length(sch)] do
#     if sch[i] < 0 then
#       out[i] := reps[-sch[i]];
#     fi;
#
#     trace  := DIGRAPHS_TraceSchreierVector(gens, sch, i);
#     out[i] := out[trace.representative];
#     word   := trace.word;
#     for w in word do
#        out[i] := OnTuples(out[i], gens[w]);
#     od;
#   od;

#   return out;
# end);

InstallMethod(DigraphAdjacencyFunction, "for a digraph by out-neighbours",
[IsDigraph], D -> {u, v} -> IsDigraphEdge(D, u, v));

InstallMethod(AsTransformation, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  if not IsFunctionalDigraph(D) then
    return fail;
  fi;
  return Transformation(Concatenation(OutNeighbours(D)));
end);

InstallMethod(DigraphNrEdges, "for a digraph", [IsDigraphByOutNeighboursRep],
function(D)
  local m;
  m := DIGRAPH_NREDGES(D);
  if IsImmutableDigraph(D) then
    SetIsEmptyDigraph(D, m = 0);
  fi;
  return m;
end);

InstallMethod(DigraphNrAdjacencies, "for a digraph",
[IsDigraphByOutNeighboursRep], DIGRAPH_NRADJACENCIES);

InstallMethod(DigraphNrAdjacenciesWithoutLoops, "for a digraph",
[IsDigraphByOutNeighboursRep], DIGRAPH_NRADJACENCIESWITHOUTLOOPS);

InstallMethod(DigraphNrLoops,
"for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local i, j, sum;
    sum := 0;
  if HasDigraphHasLoops(D) and not DigraphHasLoops(D) then
    return 0;
  else
    for i in DigraphVertices(D) do
      for j in OutNeighbours(D)[i] do
        if i = j then
          sum := sum + 1;
        fi;
      od;
    od;
  fi;
  if IsImmutableDigraph(D) then
    SetDigraphHasLoops(D, sum <> 0);
  fi;
  return sum;
end);

InstallMethod(DigraphNrLoops,
"for a digraph that knows its adjacency matrix",
[IsDigraphByOutNeighboursRep and HasAdjacencyMatrix],
function(D)
  local A, i;
  A := AdjacencyMatrix(D);
  return Sum(DigraphVertices(D), i -> A[i][i]);
end);

InstallMethod(DigraphEdges, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local out, adj, nr, i, j;
  out := EmptyPlist(DigraphNrEdges(D));
  adj := OutNeighbours(D);
  nr := 0;

  for i in DigraphVertices(D) do
    for j in adj[i] do
      nr := nr + 1;
      out[nr] := [i, j];
    od;
  od;
  return out;
end);

# Hash related attributes

InstallMethod(DigraphHash, "for a digraph", [IsDigraph], DIGRAPH_HASH);

# To make built in Orbit function use DigraphHash
InstallMethod(SparseIntKey, "for an object and digraph",
[IsObject, IsDigraph],
{coll, D} -> DigraphHash
);

# To make orb package use DigraphHash
InstallMethod(ChooseHashFunction, "for a digraph and positive integer",
[IsDigraph, IsInt],
# TODO: (reiniscirpons) remove the rank when new semigroups version gets
# released.
2,
{D, hashlen} -> rec(func := {x, data} -> 1 + (DigraphHash(x) mod data[1]),
                    data := [hashlen])
);

# attributes for digraphs . . .

InstallMethod(AsGraph, "for a digraph", [IsDigraph], Graph);

InstallMethod(DigraphVertices, "for a digraph", [IsDigraph],
D -> [1 .. DigraphNrVertices(D)]);

InstallMethod(DigraphRange,
"for an immutable digraph by out-neighbours",
[IsDigraphByOutNeighboursRep and IsImmutableDigraph],
function(D)
  if not IsBound(D!.DigraphRange) then
    DIGRAPH_SOURCE_RANGE(D);
    SetDigraphSource(D, D!.DigraphSource);
  fi;
  return D!.DigraphRange;
end);

InstallMethod(DigraphRange,
"for a mutable digraph by out-neighbours",
[IsDigraphByOutNeighboursRep and IsMutableDigraph],
D -> DIGRAPH_SOURCE_RANGE(D).DigraphRange);

InstallMethod(DigraphSource,
"for an immutable digraph by out-neighbours",
[IsDigraphByOutNeighboursRep and IsImmutableDigraph],
function(D)
  if not IsBound(D!.DigraphSource) then
    DIGRAPH_SOURCE_RANGE(D);
    SetDigraphRange(D, D!.DigraphRange);
  fi;
  return D!.DigraphSource;
end);

InstallMethod(DigraphSource,
"for a mutable digraph by out-neighbours",
[IsDigraphByOutNeighboursRep and IsMutableDigraph],
D -> DIGRAPH_SOURCE_RANGE(D).DigraphSource);

InstallMethod(InNeighbours, "for a digraph", [IsDigraphByOutNeighboursRep],
D -> DIGRAPH_IN_OUT_NBS(OutNeighbours(D)));

InstallMethod(AdjacencyMatrix, "for a digraph", [IsDigraphByOutNeighboursRep],
ADJACENCY_MATRIX);

InstallMethod(BooleanAdjacencyMatrix, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local n, nbs, mat, i, j;
  n := DigraphNrVertices(D);
  nbs := OutNeighbours(D);
  mat := List(DigraphVertices(D), x -> BlistList([1 .. n], []));
  for i in DigraphVertices(D) do
    for j in nbs[i] do
      mat[i][j] := true;
    od;
  od;
  return mat;
end);

InstallMethod(DigraphShortestDistances, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local vertices, data, sum, distances, v, u;
  if HasDIGRAPHS_ConnectivityData(D) then
    vertices := DigraphVertices(D);
    data := DIGRAPHS_ConnectivityData(D);
    sum := 0;
    for v in vertices do
      if IsBound(data[v]) then
        sum := sum + 1;
      fi;
    od;
    if sum > Int(0.9 * DigraphNrVertices(D))
        or (HasDigraphGroup(D) and not IsTrivial(DigraphGroup(D)))  then
      # adjust the constant 0.9 and possibly make a decision based on
      # how big the group is
      distances := [];
      for u in vertices do
        distances[u] := [];
        for v in vertices do
          distances[u][v] := DigraphShortestDistance(D, u, v);
        od;
      od;
      return distances;
    fi;
  fi;
  return DIGRAPH_SHORTEST_DIST(D);
end);

# returns the vertices (i.e. numbers) of <D> ordered so that there are no
# edges from <out[j]> to <out[i]> for all <i> greater than <j>.

InstallMethod(DigraphTopologicalSort, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
D -> DIGRAPH_TOPO_SORT(OutNeighbours(D)));

InstallMethod(DigraphStronglyConnectedComponents,
"for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local verts;

  if HasIsAcyclicDigraph(D) and IsAcyclicDigraph(D) then
    verts := DigraphVertices(D);
    return rec(comps := List(verts, x -> [x]), id := verts * 1);

  elif HasIsStronglyConnectedDigraph(D)
      and IsStronglyConnectedDigraph(D) then
    verts := DigraphVertices(D);
    return rec(comps := [verts * 1], id := verts * 0 + 1);
  fi;

  return GABOW_SCC(OutNeighbours(D));
end);

InstallMethod(DigraphNrStronglyConnectedComponents, "for a digraph",
[IsDigraph],
D -> Length(DigraphStronglyConnectedComponents(D).comps));

InstallMethod(DigraphConnectedComponents, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
DIGRAPH_CONNECTED_COMPONENTS);

InstallMethod(DigraphNrConnectedComponents, "for a digraph",
[IsDigraph],
D -> Length(DigraphConnectedComponents(D).comps));

InstallImmediateMethod(DigraphNrConnectedComponents,
"for a digraph with known connected components",
IsDigraph and HasDigraphConnectedComponents, 0,
D -> Length(DigraphConnectedComponents(D).comps));

InstallMethod(OutDegrees, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local adj, degs, i;
  adj := OutNeighbours(D);
  degs := EmptyPlist(DigraphNrVertices(D));
  for i in DigraphVertices(D) do
    degs[i] := Length(adj[i]);
  od;
  return degs;
end);

InstallMethod(InDegrees, "for a digraph with in neighbours",
[IsDigraph and HasInNeighbours],
2,  # to beat the method for IsDigraphByOutNeighboursRep
function(D)
  local inn, degs, i;
  inn := InNeighbours(D);
  degs := EmptyPlist(DigraphNrVertices(D));
  for i in DigraphVertices(D) do
    degs[i] := Length(inn[i]);
  od;
  return degs;
end);

InstallMethod(InDegrees, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local adj, degs, x, i;
  adj := OutNeighbours(D);
  degs := ListWithIdenticalEntries(DigraphNrVertices(D), 0);
  for x in adj do
    for i in x do
      degs[i] := degs[i] + 1;
    od;
  od;
  return degs;
end);

InstallMethod(OutDegreeSequence, "for a digraph", [IsDigraph],
function(D)
  D := ShallowCopy(OutDegrees(D));
  Sort(D, {a, b} -> b < a);
  return D;
  # return SortedList(OutDegrees(D), {a, b} -> b < a);
end);

InstallMethod(OutDegreeSequence,
"for a digraph by out-neighbours with known digraph group",
[IsDigraphByOutNeighboursRep and HasDigraphGroup],
function(D)
  local out, adj, orbs, orb;
  out := [];
  adj := OutNeighbours(D);
  orbs := DigraphOrbits(D);
  for orb in orbs do
    Append(out, [1 .. Length(orb)] * 0 + Length(adj[orb[1]]));
  od;
  Sort(out, {a, b} -> b < a);
  return out;
end);

InstallMethod(OutDegreeSet, "for a digraph", [IsDigraph],
D -> Set(ShallowCopy(OutDegrees(D))));

InstallMethod(InDegreeSequence, "for a digraph", [IsDigraph],
function(D)
  D := ShallowCopy(InDegrees(D));
  Sort(D, {a, b} -> b < a);
  return D;
  # return SortedList(OutDegrees(D), {a, b} -> b < a);
end);

InstallMethod(InDegreeSequence,
"for a digraph with known digraph group and in-neighbours",
[IsDigraph and HasDigraphGroup and HasInNeighbours],
function(D)
  local out, adj, orbs, orb;
  out := [];
  adj := InNeighbours(D);
  orbs := DigraphOrbits(D);
  for orb in orbs do
    Append(out, [1 .. Length(orb)] * 0 + Length(adj[orb[1]]));
  od;
  Sort(out, {a, b} -> b < a);
  return out;
end);

InstallMethod(InDegreeSet, "for a digraph", [IsDigraph],
D -> Set(ShallowCopy(InDegrees(D))));

InstallMethod(DigraphSources, "for a digraph with in-degrees",
[IsDigraph and HasInDegrees], 3,
function(D)
  local degs;
  degs := InDegrees(D);
  return Filtered(DigraphVertices(D), x -> degs[x] = 0);
end);

InstallMethod(DigraphSources, "for a digraph with in-neighbours",
[IsDigraph and HasInNeighbours],
2,  # to beat the method for IsDigraphByOutNeighboursRep
function(D)
  local inn, sources, count, i;

  inn := InNeighbours(D);
  sources := EmptyPlist(DigraphNrVertices(D));
  count := 0;
  for i in DigraphVertices(D) do
    if IsEmpty(inn[i]) then
      count := count + 1;
      sources[count] := i;
    fi;
  od;
  ShrinkAllocationPlist(sources);
  return sources;
end);

InstallMethod(DigraphSources, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local out, seen, tmp, next, v;
  out  := OutNeighbours(D);
  seen := BlistList(DigraphVertices(D), []);
  for next in out do
    for v in next do
      seen[v] := true;
    od;
  od;
  # FIXME use FlipBlist (when available)
  tmp  := BlistList(DigraphVertices(D), DigraphVertices(D));
  SubtractBlist(tmp, seen);
  return ListBlist(DigraphVertices(D), tmp);
end);

InstallMethod(DigraphSinks, "for a digraph with out-degrees",
[IsDigraph and HasOutDegrees],
2,  # to beat the method for IsDigraphByOutNeighboursRep
function(D)
  local degs;
  degs := OutDegrees(D);
  return Filtered(DigraphVertices(D), x -> degs[x] = 0);
end);

InstallMethod(DigraphSinks, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local out, sinks, count, i;

  out   := OutNeighbours(D);
  sinks := [];
  count := 0;
  for i in DigraphVertices(D) do
    if IsEmpty(out[i]) then
      count := count + 1;
      sinks[count] := i;
    fi;
  od;
  return sinks;
end);

InstallMethod(DigraphPeriod, "for a digraph", [IsDigraphByOutNeighboursRep],
function(D)
  local comps, out, deg, nrvisited, period, stack, len, depth, current,
        olddepth, i;

  if HasIsAcyclicDigraph(D) and IsAcyclicDigraph(D) then
    return 0;
  fi;

  comps := DigraphStronglyConnectedComponents(D).comps;
  out := OutNeighbours(D);
  deg := OutDegrees(D);

  nrvisited := [1 .. Length(DigraphVertices(D))] * 0;
  period := 0;

  for i in [1 .. Length(comps)] do
    stack := [comps[i][1]];
    len := 1;
    depth := EmptyPlist(Length(DigraphVertices(D)));
    depth[comps[i][1]] := 1;
    while len <> 0 do
      current := stack[len];
      if nrvisited[current] = deg[current] then
        len := len - 1;
      else
        nrvisited[current] := nrvisited[current] + 1;
        len := len + 1;
        stack[len] := out[current][nrvisited[current]];
        olddepth := depth[current];
        if IsBound(depth[stack[len]]) then
          period := GcdInt(period, depth[stack[len]] - olddepth - 1);
          if period = 1 then
            return period;
          fi;
        else
          depth[stack[len]] := olddepth + 1;
        fi;
      fi;
    od;
  od;

  if period = 0 and IsImmutableDigraph(D) then
    SetIsAcyclicDigraph(D, true);
  fi;

  return period;
end);

InstallMethod(DIGRAPHS_ConnectivityData, "for a digraph", [IsDigraph],
D -> EmptyPlist(0));

BindGlobal("DIGRAPH_ConnectivityDataForVertex",
function(D, v)
  local data, out_nbs, record, orbnum, reps, i, next, laynum, localGirth,
        layers, sum, localParameters, nprev, nhere, nnext, lnum, localDiameter,
        layerNumbers, x, y, tree, expand, stab, edges, edge;

  data := DIGRAPHS_ConnectivityData(D);
  if IsBound(data[v]) then
    return data[v];
  fi;

  expand := false;
  if HasDigraphGroup(D) then
    stab            := DigraphStabilizer(D, v);
    record          := DIGRAPHS_Orbits(stab, DigraphVertices(D));
    orbnum          := record.lookup;
    reps            := List(record.orbits, Representative);
    if Length(record.orbits) < DigraphNrVertices(D) then
        expand := true;
    fi;
  else
    orbnum          := [1 .. DigraphNrVertices(D)];
    reps            := [1 .. DigraphNrVertices(D)];
  fi;
  out_nbs         := OutNeighbours(D);
  i               := 1;
  next            := [orbnum[v]];
  laynum          := ListWithIdenticalEntries(Length(reps), 0);
  laynum[next[1]] := 1;
  tree            := ListWithIdenticalEntries(DigraphNrVertices(D), 0);
  tree[v]         := -1;
  localGirth      := -1;
  layers          := [next];
  sum             := 1;
  localParameters := [];

  # localDiameter is the length of the longest shortest path starting at v
  #
  # localParameters is a list of 3-tuples [a_{i - 1}, b_{i - 1}, c_{i - 1}] for
  # each i between 1 and localDiameter where c_i (respectively a_i and b_i) is
  # the number of vertices at distance i − 1 (respectively i and i + 1) from v
  # that are adjacent to a vertex w at distance i from v.

  # <tree> gives a shortest path spanning tree rooted at <v> and is used by
  # the ShortestPathSpanningTree method.

  while Length(next) > 0 do
    next := [];
    for x in layers[i] do
      nprev := 0;
      nhere := 0;
      nnext := 0;
      for y in out_nbs[reps[x]] do
        lnum := laynum[orbnum[y]];
        if i > 1 and lnum = i - 1 then
          nprev := nprev + 1;
        elif lnum = i then
          nhere := nhere + 1;
        elif lnum = i + 1 then
          nnext := nnext + 1;
        elif lnum = 0 then
          AddSet(next, orbnum[y]);
          nnext := nnext + 1;
          laynum[orbnum[y]] := i + 1;
          if not expand then
            tree[y] := reps[x];
          else
            edges := Orbit(stab, [reps[x], y], OnTuples);
            for edge in edges do
              if tree[edge[2]] = 0 or tree[edge[2]] > edge[1] then
                tree[edge[2]] := edge[1];
              fi;
            od;
          fi;
        fi;
      od;
      if (localGirth = -1 or localGirth = 2 * i - 1) and nprev > 1 then
        localGirth := 2 * (i - 1);
      fi;
      if localGirth = -1 and nhere > 0 then
        localGirth := 2 * i - 1;
      fi;
      if not IsBound(localParameters[i]) then
         localParameters[i] := [nprev, nhere, nnext];
      else
         if nprev <> localParameters[i][1] then
            localParameters[i][1] := -1;
         fi;
         if nhere <> localParameters[i][2] then
            localParameters[i][2] := -1;
         fi;
         if nnext <> localParameters[i][3] then
            localParameters[i][3] := -1;
         fi;
      fi;
    od;
    if Length(next) > 0 then
      i := i + 1;
      layers[i] := next;
      sum := sum + Length(next);
    fi;
  od;
  if sum = Length(reps) then
     localDiameter := Length(layers) - 1;
  else
     localDiameter := -1;
  fi;

  layerNumbers := [];
  for i in DigraphVertices(D) do
     layerNumbers[i] := laynum[orbnum[i]];
  od;
  data[v] := rec(layerNumbers    := layerNumbers,
                 localDiameter   := localDiameter,
                 localGirth      := localGirth,
                 localParameters := localParameters,
                 layers          := layers,
                 spstree         := tree);
  return data[v];
end);

BindGlobal("DIGRAPHS_DiameterAndUndirectedGirth",
function(D)
  local outer_reps, diameter, girth, v, record, localGirth,
        localDiameter, i;

  # This function attempts to find the diameter and undirected girth of a given
  # D, using its DigraphGroup.  For some digraphs, the main algorithm will
  # not produce a sensible answer, so there are checks at the start and end to
  # alter the answer for the diameter/girth if necessary.  This function is
  # called, if appropriate, by DigraphDiameter and DigraphUndirectedGirth.

  if DigraphHasNoVertices(D) and IsImmutableDigraph(D) then
    SetDigraphDiameter(D, fail);
    SetDigraphUndirectedGirth(D, infinity);
    return rec(diameter := fail, girth := infinity);
  fi;

  # TODO improve this, really check if the complexity is better with the group
  # or without, or if the group is not known, but the number of vertices makes
  # the usual algorithm impossible.

  outer_reps := DigraphOrbitReps(D);
  diameter   := 0;
  girth      := infinity;

  for i in [1 .. Length(outer_reps)] do
    v := outer_reps[i];
    record     := DIGRAPH_ConnectivityDataForVertex(D, v);
    localGirth := record.localGirth;
    localDiameter := record.localDiameter;

    if localDiameter > diameter then
      diameter := localDiameter;
    fi;

    if localGirth <> -1 and localGirth < girth then
      girth := localGirth;
    fi;
  od;

  # Checks to ensure both components are valid
  if not IsStronglyConnectedDigraph(D) then
    diameter := fail;
  fi;
  if DigraphHasLoops(D) then
    girth := 1;
  elif IsMultiDigraph(D) then
    girth := 2;
  fi;

  if IsImmutableDigraph(D) then
    SetDigraphDiameter(D, diameter);
    SetDigraphUndirectedGirth(D, girth);
  fi;
  return rec(diameter := diameter, girth := girth);
end);

InstallMethod(DigraphDiameter, "for a digraph", [IsDigraph],
function(D)
  if not IsStronglyConnectedDigraph(D) then
    # Diameter undefined
    return fail;
  elif HasDigraphGroup(D) and Size(DigraphGroup(D)) > 1 then
    # Use the group to calculate the diameter
    return DIGRAPHS_DiameterAndUndirectedGirth(D).diameter;
  fi;
  # Use the C function
  return DIGRAPH_DIAMETER(D);
end);

InstallMethod(DigraphUndirectedGirth, "for a digraph", [IsDigraph],
function(D)
  # This is only defined on undirected graphs (i.e. symmetric digraphs)
  if not IsSymmetricDigraph(D) then
    ErrorNoReturn("the argument <D> must be a symmetric digraph,");
  elif DigraphHasLoops(D) then
    # A loop is a cycle of length 1
    return 1;
  elif IsMultiDigraph(D) then
    # A pair of multiple edges is a cycle of length 2
    return 2;
  fi;
  # Otherwise D is simple
  return DIGRAPHS_DiameterAndUndirectedGirth(D).girth;
end);

InstallMethod(DigraphGirth, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local verts, girth, out, dist, i, j;
  if DigraphHasLoops(D) then
    return 1;
  fi;
  # Only consider one vertex from each orbit
  if HasDigraphGroup(D) and not IsTrivial(DigraphGroup(D)) then
    verts := DigraphOrbitReps(D);
  else
    verts := DigraphVertices(D);
  fi;
  girth := infinity;
  out := OutNeighbours(D);
  for i in verts do
    for j in out[i] do
      dist := DigraphShortestDistance(D, j, i);
      # distance [j,i] + 1 equals the cycle length
      if dist <> fail and dist + 1 < girth then
        girth := dist + 1;
        if girth = 2 then
          return girth;
        fi;
      fi;
    od;
  od;
  return girth;
end);

InstallMethod(DigraphOddGirth, "for a digraph",
[IsDigraph],
function(digraph)
  local comps, girth, oddgirth, A, B, gr, k, comp;

  if IsAcyclicDigraph(digraph) then
    return infinity;
  elif IsOddInt(DigraphGirth(digraph)) then
    # No need to check girth isn't infinity, as we have
    # that digraph is not acyclic.
    return DigraphGirth(digraph);
  fi;
  comps := DigraphStronglyConnectedComponents(digraph).comps;
  if Length(comps) > 1 and IsMutableDigraph(digraph) then
    # Necessary because InducedSubdigraph alters mutable args
    digraph := DigraphImmutableCopy(digraph);
  fi;
  oddgirth := infinity;
  for comp in comps do
    if Length(comps) > 1 then  # i.e. if not IsStronglyConnectedDigraph(digraph)
      gr := InducedSubdigraph(digraph, comp);
    else
      gr := digraph;
      # If digraph is strongly connected, then we needn't
      # induce the subdigraph of its strongly connected comp.
    fi;
    if not IsAcyclicDigraph(gr) then
      girth := DigraphGirth(gr);
      if IsOddInt(girth) and girth < oddgirth then
        oddgirth := girth;
      else
        k := girth - 1;
        A := AdjacencyMatrix(gr) ^ k;
        B := AdjacencyMatrix(gr) ^ 2;
        while k <= DigraphNrEdges(gr) + 2 and k < DigraphNrVertices(gr)
            and k < oddgirth - 2 do
          A := A * B;
          k := k + 2;
          if Trace(A) <> 0 then
            # It suffices to find the trace as the entries of A are positive.
            oddgirth := k;
          fi;
        od;
      fi;
    fi;
  od;

  return oddgirth;
end);

InstallMethod(DigraphLongestSimpleCircuit, "for a digraph",
[IsDigraph],
function(D)
  local circs, lens, max;
  if IsAcyclicDigraph(D) then
    return fail;
  fi;
  circs := DigraphAllSimpleCircuits(D);
  lens := List(circs, Length);
  max := Maximum(lens);
  return circs[Position(lens, max)];
end);

InstallMethod(DigraphAllSimpleCircuits, "for a digraph by out-neighbours",
[IsDigraphByOutNeighboursRep],
function(D)
  local UNBLOCK, CIRCUIT, out, stack, endofstack, C, scc, n, blocked, B,
  c_comp, comp, s, loops, i, d_labels, c_labels;

  if IsEmptyDigraph(D) then
    return [];
  fi;

  UNBLOCK := function(u)
    local w;
    blocked[u] := false;
    while not IsEmpty(B[u]) do
      w := B[u][1];
      Remove(B[u], 1);
      if blocked[w] then
        UNBLOCK(w);
      fi;
    od;
  end;

  CIRCUIT := function(v, component)
    local f, buffer, dummy, w;

    f := false;
    endofstack := endofstack + 1;
    stack[endofstack] := v;
    blocked[v] := true;

    for w in OutNeighboursOfVertex(component, v) do
      if w = 1 then
        buffer := stack{[1 .. endofstack]};
        Add(out, DigraphVertexLabels(component){buffer});
        f := true;
      elif blocked[w] = false then
        dummy := CIRCUIT(w, component);
        if dummy then
          f := true;
        fi;
      fi;
    od;

    if f then
      UNBLOCK(v);
    else
      for w in OutNeighboursOfVertex(component, v) do
        if not w in B[w] then
          Add(B[w], v);
        fi;
      od;
    fi;

    endofstack := endofstack - 1;
    return f;
  end;

  out := [];
  stack := [];
  endofstack := 0;

  # Reduce the D, remove loops, and store the correct vertex labels
  C := DigraphRemoveLoops(ReducedDigraph(DigraphMutableCopyIfMutable(D)));
  MakeImmutable(C);
  if HaveVertexLabelsBeenAssigned(D)
      and DigraphVertexLabels(D) <> DigraphVertices(D) then
    # We require the labels of the digraph <C> to be the original nodes in <D>
    # (this is used in CIRCUIT above). If <D> has other vertex labels, then
    # these are copied into <C> by the functions above, and aren't then
    # necessarily the original nodes in <D>.
    d_labels := DigraphVertexLabels(D);
    c_labels := List(DigraphVertexLabels(C), x -> Position(d_labels, x));
    SetDigraphVertexLabels(C, c_labels);
  fi;

  # Strongly connected components of the reduced graph
  scc := DigraphStronglyConnectedComponents(C);

  # B and blocked only need to be as long as the longest connected component
  n := Maximum(List(scc.comps, Length));
  blocked := BlistList([1 .. n], []);
  B := List([1 .. n], x -> []);

  # Perform algorithm once per connected component of the whole digraph
  for c_comp in scc.comps do
    n := Length(c_comp);
    if n = 1 then
      continue;
    fi;
    c_comp := InducedSubdigraph(C, c_comp);  # C is definitely immutable
    comp := c_comp;
    s := 1;
    while s < n do
      if s <> 1 then
        comp := InducedSubdigraph(c_comp, [s .. n]);
        comp := InducedSubdigraph(comp,
                                  DigraphStronglyConnectedComponent(comp, 1));
      fi;

      if not IsEmptyDigraph(comp) then
        # TODO would it be faster/better to create blocked as a new BlistList?
        # Are these things already going to be initialised anyway?
        for i in DigraphVertices(comp) do
          blocked[i] := false;
          B[i] := [];
        od;
        CIRCUIT(1, comp);
      fi;
      s := s + 1;
    od;
  od;
  loops := List(DigraphLoops(D), x -> [x]);
  return Concatenation(loops, out);
end);

# Compute all undirected simple circuits by filtering the output
# of DigraphAllSimpleCircuits
InstallMethod(DigraphAllUndirectedSimpleCircuits, "for a digraph", [IsDigraph],
function(D)
  local digraph, cycles, keep, cycle, cycleRev;
  digraph := DigraphSymmetricClosure(
      DigraphRemoveAllMultipleEdges(DigraphMutableCopyIfMutable(D)));
  cycles := Filtered(DigraphAllSimpleCircuits(digraph),
    c -> Length(c) > 2 or Length(c) = 1);
  keep := HashSet();
  for cycle in cycles do
    cycleRev := [cycle[1]];
    Append(cycleRev, Reversed(cycle{[2 .. Length(cycle)]}));
    if (not cycle in keep and not cycleRev in keep) or Length(cycle) = 1 then
      AddSet(keep, cycle);
    fi;
  od;
  return Set(keep);
end);

# Compute all chordless cycles for a given symmetric digraph
# Algorithm based on https://arxiv.org/pdf/1404.7610
InstallMethod(DigraphAllChordlessCyclesOfMaximalLength,
"for a digraph and an integer", [IsDigraph, IsInt],
function(D, maxLength)
  local BlockNeighbours, UnblockNeighbours,
  Triplets, CCExtension, digraph, temp, T, C, blocked, triple;

    if IsEmptyDigraph(D) then
        return [];
    fi;

    BlockNeighbours := function(digraph, v, blocked)
        local u;
        for u in OutNeighboursOfVertex(digraph, v) do
            blocked[u] := blocked[u] + 1;
        od;
        return blocked;
    end;

    UnblockNeighbours := function(digraph, v, blocked)
        local u;
        for u in OutNeighboursOfVertex(digraph, v) do
            if blocked[u] > 0 then
                blocked[u] := blocked[u] - 1;
            fi;
        od;
        return blocked;
    end;

    # Computes all possible triplets
    Triplets := function(digraph)
        local T, C, u, pair, x, y, labels;
        T := [];
        C := [];
        for u in DigraphVertices(digraph) do
            for pair in Combinations(OutNeighboursOfVertex(digraph, u), 2) do
                x := pair[1];
                y := pair[2];
                labels := DigraphVertexLabels(digraph);
                if labels[u] < labels[x] and labels[x] < labels[y] then
                    if not IsDigraphEdge(digraph, x, y) then
                        Add(T, [x, u, y]);
                    else
                        Add(C, [x, u, y]);
                    fi;
                elif labels[u] < labels[y] and labels[y] < labels[x] then
                    if not IsDigraphEdge(digraph, x, y) then
                        Add(T, [y, u, x]);
                    else
                        Add(C, [y, u, x]);
                    fi;
                fi;
            od;
        od;
        return [T, C];
    end;

    # Extends a given chordless path if possible
    CCExtension := function(digraph, path, C, key, blocked)
        local v, extendedPath, data;
        blocked := BlockNeighbours(digraph, path[Length(path)], blocked);
        for v in OutNeighboursOfVertex(digraph, path[Length(path)]) do
            if DigraphVertexLabel(digraph, v) > key and blocked[v] = 1
              and Length(path) < maxLength then
                extendedPath := Concatenation(path, [v]);
                if IsDigraphEdge(digraph, v, path[1]) then
                    Add(C, extendedPath);
                else
                    data := CCExtension(digraph, extendedPath, C, key, blocked);
                    C := data[1];
                    blocked := data[2];
                fi;
            fi;
        od;
        blocked := UnblockNeighbours(digraph, path[Length(path)], blocked);
        return [C, blocked];
    end;

    digraph := DigraphMutableCopy(D);
    DigraphSymmetricClosure(DigraphRemoveLoops(
                 DigraphRemoveAllMultipleEdges(digraph)));
    MakeImmutable(digraph);
    SetDigraphVertexLabels(digraph,
                 Reversed(DigraphDegeneracyOrdering(digraph)));

    temp := Triplets(digraph);
    T := temp[1];
    C := temp[2];
    blocked := List(DigraphVertices(digraph), i -> 0);
    while T <> [] do
        triple := Remove(T);
        blocked := BlockNeighbours(digraph, triple[2], blocked);
        temp := CCExtension(digraph, triple, C,
                              DigraphVertexLabel(digraph, triple[2]), blocked);
        C := temp[1];
        blocked := temp[2];
        blocked := UnblockNeighbours(digraph, triple[2], blocked);
    od;
    return C;
end);

InstallMethod(DigraphAllChordlessCycles, "for a digraph",
[IsDigraph],
D -> DigraphAllChordlessCyclesOfMaximalLength(D, INTOBJ_MAX));

# Compute for a given rotation system the facial walks
InstallMethod(FacialWalks, "for a digraph and a list",
[IsDigraph, IsDenseList],
function(D, rotationSystem)

    local FacialWalk, facialWalks, remEdges, cycle;

    if not IsEulerianDigraph(D) then
        Error("the 1st argument (digraph <D>) must be Eulerian, but it is not");
    fi;

    if Length(rotationSystem) <> DigraphNrVertices(D) then
        Error("the 2nd argument (list <rotationSystem>) is not a rotation ",
              "system for the 1st argument (digraph <D>), expected a ",
              "dense list of length ", DigraphNrVertices(D),
              "but found dense list of length ", Length(rotationSystem));
    fi;

    if Difference(Union(rotationSystem), DigraphVertices(D))
       <> [] then
        Error("the 2nd argument (dense list <rotationSystem>) is not ",
              "a rotation system for the 1st argument (digraph <D>), ",
              "expected the union to be ", DigraphVertices(D), " but found ",
              Union(rotationSystem));
    fi;

    # computes a facial cycles starting with the edge 'startEdge'
    FacialWalk := function(rotationSystem, startEdge)
        local startVertex, preVertex, actVertex, cycle, nextVertex, pos;

        startVertex := startEdge[1];
        actVertex := startEdge[2];
        preVertex := startVertex;

        cycle := [startVertex, actVertex];

        nextVertex := 0;  # just an initialization
        while true do
            pos := Position(rotationSystem[actVertex], preVertex);

            if pos < Length(rotationSystem[actVertex]) then
                nextVertex := rotationSystem[actVertex][pos + 1];
            else
                nextVertex := rotationSystem[actVertex][1];
            fi;
            if nextVertex <> startEdge[2] or actVertex <> startVertex then
                Add(cycle, nextVertex);
                Remove(remEdges, Position(remEdges, [preVertex, actVertex]));
                preVertex := actVertex;
                actVertex := nextVertex;
            else
                break;
            fi;
        od;
        Remove(remEdges, Position(remEdges, [preVertex, startVertex]));
        # Remove the last vertex, otherwise otherwise
        # the start vertex is contained twice
        Remove(cycle);
        return cycle;
    end;

    D := DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(
        DigraphMutableCopyIfMutable(D)));

    facialWalks := [];
    remEdges := ShallowCopy(DigraphEdges(D));

    while remEdges <> [] do
        cycle := FacialWalk(rotationSystem, remEdges[1]);
        Add(facialWalks, cycle);
    od;
    return facialWalks;
end);

# Computes the minimal cyclic edge cut of connected cubic graphs with at
# least 8 vertices based on the paper "An Algorithm for Cyclic Edge Connectivity
# of Cubic Graphs"
InstallMethod(MinimalCyclicEdgeCut, "for a digraph", [IsDigraph],
function(digraph)
  local girth, cut, cutsize, vertex, treedepth, paths, treev, treew,
  minimal_cycle, v, w, FullTree, FindCut, FindPath, e;

  # Compute a full tree of the given depth centered at the given vertex
  FullTree := function(digraph, vertex, depth)
    local result, i, lastTree, node;
    result := Set([vertex]);
    for i in [1 .. depth] do
      lastTree := ShallowCopy(result);
      for node in lastTree do
          UniteSet(result, OutNeighboursOfVertex(digraph, node));
      od;
    od;
    return result;
  end;

  # Compute a maximal amount of paths between the two distinct sets of
  # nodes treev and treew by using a variation of the Ford-Fulkerson algorithm
  FindPath := function(digraph, treev, treew)
    local digraphCopy, pathlist, newPathFound, node1, node2, path, i;

    digraphCopy := DigraphMutableCopy(digraph);
    pathlist := [];
    newPathFound := true;
    while newPathFound do
      newPathFound := false;
      for node1 in treev do
        for node2 in treew do
          path := DigraphShortestPath(digraphCopy, node1, node2);

          if path <> fail then
            for i in [1 .. (Length(path[1]) - 1)] do
              # remove edges corresponding to the current path
              digraphCopy := DigraphRemoveEdge(digraphCopy,
                [path[1][i], path[1][i + 1]]);
              # add backward edges, if they are missing
              if not [path[1][i + 1], path[1][i]] in
                    DigraphEdges(digraphCopy) then
                  digraphCopy := DigraphAddEdge(digraphCopy,
                    [path[1][i + 1], path[1][i]]);
              fi;
            od;
            Append(pathlist, [path[1]]);
            newPathFound := true;
            break;
          fi;
        od;
      od;
    od;
    return pathlist;
  end;

  # Finds a cut that disconnects the node sets treev and treew in
  # the given digraph by using the paths found in FindPath
  FindCut := function(digraph, treev, treew, allPaths)
    local pathByEdges, path, edgeList, i, cutEdges, edge, component1,
    component2, edgeSet, digraphCopy, permutations, nodeSet,
    v, pathInducedSubgraph;

    # Convert the paths into the list of corresponding edges
    pathByEdges := [];
    for path in allPaths do
      edgeList := [];
      for i in [1 .. Length(path) - 1] do
          Append(edgeList, [[path[i], path[i + 1]]]);
      od;
      Append(pathByEdges, [edgeList]);
    od;

    nodeSet := Set([]);
    for path in allPaths do
      for v in path do
        AddSet(nodeSet, v);
      od;
    od;

    edgeSet := Set([]);
    for v in DigraphVertices(digraph) do
      for w in OutNeighboursOfVertex(digraph, v) do
        UniteSet(edgeSet, [[v, w]]);
        UniteSet(edgeSet, [[w, v]]);
      od;
    od;

    # We can find a cut by removing one specific edge from every path,
--> --------------------

--> maximum size reached

--> --------------------

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