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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.
##
#############################################################################
##
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
--> --------------------
[ 0.68Quellennavigators
Projekt
]
|
2026-03-28
|
