```-- Copyright (c) David Amos, 2009. All rights reserved.

module Math.Combinatorics.GraphAuts where

import qualified Data.List as L
import qualified Data.Map as M
import qualified Data.Set as S
import Data.Maybe

import Math.Common.ListSet
import Math.Combinatorics.Graph
-- import Math.Combinatorics.StronglyRegularGraph
-- import Math.Combinatorics.Hypergraph -- can't import this, creates circular dependency
import Math.Algebra.Group.PermutationGroup
import Math.Algebra.Group.SchreierSims as SS

-- The code for finding automorphisms - "graphAuts" - follows later on in file

-- TRANSITIVITY PROPERTIES OF GRAPHS

-- |A graph is vertex-transitive if its automorphism group acts transitively on the vertices. Thus, given any two distinct vertices, there is an automorphism mapping one to the other.
isVertexTransitive :: (Ord t) => Graph t -> Bool
isVertexTransitive (G [] []) = True -- null graph is trivially vertex transitive
isVertexTransitive g@(G (v:vs) es) = orbitV auts v == v:vs where
auts = graphAuts g

-- |A graph is edge-transitive if its automorphism group acts transitively on the edges. Thus, given any two distinct edges, there is an automorphism mapping one to the other.
isEdgeTransitive :: (Ord t) => Graph t -> Bool
isEdgeTransitive (G _ []) = True
isEdgeTransitive g@(G vs (e:es)) = orbitE auts e == e:es where
auts = graphAuts g

arc ->^ g = map (.^ g) arc
-- unlike edges/blocks, arcs are directed, so the action on them does not sort

-- Godsil & Royle 59-60
-- |A graph is arc-transitive (or flag-transitive) if its automorphism group acts transitively on arcs. (An arc is an ordered pair of adjacent vertices.)
isArcTransitive :: (Ord t) => Graph t -> Bool
isArcTransitive (G _ []) = True -- empty graphs are trivially arc transitive
isArcTransitive g@(G vs es) = orbit (->^) a auts == a:as where
-- isArcTransitive g@(G vs es) = closure [a] [ ->^ h | h <- auts] == a:as where
a:as = L.sort \$ es ++ map reverse es
auts = graphAuts g

isArcTransitive' g@(G (v:vs) es) =
orbitP auts v == v:vs && -- isVertexTransitive g
orbitP stab n == n:ns
where auts = graphAuts g
stab = dropWhile (\p -> v .^ p /= v) auts -- we know that graphAuts are returned in this order
n:ns = nbrs g v

-- execution time of both of the above is dominated by the time to calculate the graph auts, so their performance is similar

-- then k n, kb n n, q n, other platonic solids, petersen graph, heawood graph, pappus graph, desargues graph are all arc-transitive

-- find arcs of length l from x using dfs - results returned in order
-- an arc is a sequence of vertices connected by edges, no doubling back, but self-crossings allowed
findArcs g@(G vs es) x l = map reverse \$ dfs [ ([x],0) ] where
dfs ( (z1:z2:zs,l') : nodes)
| l == l'   = (z1:z2:zs) : dfs nodes
| otherwise = dfs \$ [(w:z1:z2:zs,l'+1) | w <- nbrs g z1, w /= z2] ++ nodes
dfs ( ([z],l') : nodes)
| l == l'   = [z] : dfs nodes
| otherwise = dfs \$ [([w,z],l'+1) | w <- nbrs g z] ++ nodes
dfs [] = []

-- note that a graph with triangles can't be 3-arc transitive, etc, because an aut can't map a self-crossing arc to a non-self-crossing arc

-- |A graph is n-arc-transitive is its automorphism group is transitive on n-arcs. (An n-arc is an ordered sequence (v0,...,vn) of adjacent vertices, with crossings allowed but not doubling back.)
isnArcTransitive :: (Ord t) => Int -> Graph t -> Bool
isnArcTransitive _ (G [] []) = True
isnArcTransitive n g@(G (v:vs) es) =
orbitP auts v == v:vs && -- isVertexTransitive g
orbit (->^) a stab == a:as
-- closure [a] [ ->^ h | h <- stab] == a:as
where auts = graphAuts g
stab = dropWhile (\p -> v .^ p /= v) auts -- we know that graphAuts are returned in this order
a:as = findArcs g v n

is2ArcTransitive :: (Ord t) => Graph t -> Bool
is2ArcTransitive g = isnArcTransitive 2 g

is3ArcTransitive :: (Ord t) => Graph t -> Bool
is3ArcTransitive g = isnArcTransitive 3 g

-- Godsil & Royle 66-7
-- |A graph is distance transitive if given any two ordered pairs of vertices (u,u') and (v,v') with d(u,u') == d(v,v'),
-- there is an automorphism of the graph that takes (u,u') to (v,v')
isDistanceTransitive :: (Ord t) => Graph t -> Bool
isDistanceTransitive (G [] []) = True
isDistanceTransitive g@(G (v:vs) es)
| isConnected g =
orbitP auts v == v:vs && -- isVertexTransitive g
length stabOrbits == diameter g + 1 -- the orbits under the stabiliser of v coincide with the distance partition from v
| otherwise = error "isDistanceTransitive: only defined for connected graphs"
where auts = graphAuts g
stab = dropWhile (\p -> v .^ p /= v) auts -- we know that graphAuts are returned in this order
stabOrbits = let os = orbits stab in os ++ map (:[]) ((v:vs) L.\\ concat os) -- include fixed point orbits

-- GRAPH AUTOMORPHISMS

-- !! Note, in the literature the following is just called the intersection of two partitions
-- !! Refinement actually refers to the process of refining to an equitable partition

-- refine one partition by another
refine p1 p2 = filter (not . null) \$ refine' p1 p2
-- Refinement preserves ordering within cells but not between cells
-- eg the cell [1,2,3,4] could be refined to [2,4],[1,3]

-- refine, but leaving null cells in
-- we use this in the graphAuts functions when comparing two refinements to check that they split in the same way
refine' p1 p2 = concat [ [c1 `intersect` c2 | c2 <- p2] | c1 <- p1]

isGraphAut (G vs es) h = all (`S.member` es') [e -^ h | e <- es]
where es' = S.fromList es
-- this works best on sparse graphs, where p(edge) < 1/2
-- if p(edge) > 1/2, it would be better to test on the complement of the graph

-- Calculate a map consisting of neighbour lists for each vertex in the graph
-- If a vertex has no neighbours then it is left out of the map
adjLists' (M.insertWith' (flip (++)) v [u] \$ M.insertWith' (flip (++)) u [v] nbrs) es

-- ALTERNATIVE VERSIONS OF GRAPH AUTS
-- (showing how we got to the final version)

-- return all graph automorphisms, using naive depth first search
graphAuts1 (G vs es) = dfs [] vs vs
where dfs xys (x:xs) ys =
concat [dfs ((x,y):xys) xs (L.delete y ys) | y <- ys, isCompatible (x,y) xys]
dfs xys [] [] = [fromPairs xys]
isCompatible (x,y) xys = and [([x',x] `S.member` es') == (L.sort [y,y'] `S.member` es') | (x',y') <- xys]
es' = S.fromList es

-- return generators for graph automorphisms
-- (using Lemma 9.1.1 from Seress p203 to prune the search tree)
graphAuts2 (G vs es) = graphAuts' [] vs
where graphAuts' us (v:vs) =
let uus = zip us us
in concat [take 1 \$ dfs ((v,w):uus) vs (v : L.delete w vs) | w <- vs, isCompatible (v,w) uus]
++ graphAuts' (v:us) vs
-- stab us == transversal for stab (v:us) ++ stab (v:us)  (generators thereof)
graphAuts' _ [] = [] -- we're not interested in finding the identity element
dfs xys (x:xs) ys =
concat [dfs ((x,y):xys) xs (L.delete y ys) | y <- ys, isCompatible (x,y) xys]
dfs xys [] [] = [fromPairs xys]
isCompatible (x,y) xys = and [([x',x] `S.member` es') == (L.sort [y,y'] `S.member` es') | (x',y') <- xys]
es' = S.fromList es

-- Now using distance partitions
-- Note that because of the use of distance partitions, this is only valid for connected graphs
graphAuts3 g@(G vs es) = graphAuts' [] [vs] where
graphAuts' us ((x:ys):pt) =
let px = refine' (ys : pt) (dps M.! x)
p y = refine' ((x : L.delete y ys) : pt) (dps M.! y)
uus = zip us us
p' = L.sort \$ filter (not . null) \$ px
in concat [take 1 \$ dfs ((x,y):uus) px (p y) | y <- ys]
++ graphAuts' (x:us) p'
graphAuts' us ([]:pt) = graphAuts' us pt
graphAuts' _ [] = []
dfs xys p1 p2
| map length p1 /= map length p2 = []
| otherwise =
let p1' = filter (not . null) p1
p2' = filter (not . null) p2
in if all isSingleton p1'
then let xys' = xys ++ zip (concat p1') (concat p2')
in if isCompatible xys' then [fromPairs' xys'] else []
-- we shortcut the search when we have all singletons, so must check isCompatible to avoid false positives
else let (x:xs):p1'' = p1'
ys:p2'' = p2'
in concat [dfs ((x,y):xys)
(refine' (xs : p1'') (dps M.! x))
(refine' ((L.delete y ys):p2'') (dps M.! y))
| y <- ys]
isCompatible xys = and [([x,x'] `S.member` es') == (L.sort [y,y'] `S.member` es') | (x,y) <- xys, (x',y') <- xys, x < x']
dps = M.fromList [(v, distancePartition g v) | v <- vs]
es' = S.fromList es

isSingleton [_] = True
isSingleton _ = False

-- Now we try to use generators we've already found at a given level to save us having to look for others
-- For example, if we have found (1 2)(3 4) and (1 3 2), then we don't need to look for something taking 1 -> 4
graphAuts4 g@(G vs es) = graphAuts' [] [vs] where
graphAuts' us p@((x:ys):pt) =
-- let p' = L.sort \$ filter (not . null) \$ refine' (ys:pt) (dps M.! x)
let p' = L.sort \$ refine (ys:pt) (dps M.! x)
in level us p x ys []
++ graphAuts' (x:us) p'
graphAuts' us ([]:pt) = graphAuts' us pt
graphAuts' _ [] = []
level us p@(ph:pt) x (y:ys) hs =
let px = refine' (L.delete x ph : pt) (dps M.! x)
py = refine' (L.delete y ph : pt) (dps M.! y)
uus = zip us us
in case dfs ((x,y):uus) px py of
[]  -> level us p x ys hs
h:_ -> let hs' = h:hs in h : level us p x (ys L.\\ (x .^^ hs')) hs'
level _ _ _ [] _ = []
dfs xys p1 p2
| map length p1 /= map length p2 = []
| otherwise =
let p1' = filter (not . null) p1
p2' = filter (not . null) p2
in if all isSingleton p1'
then let xys' = xys ++ zip (concat p1') (concat p2')
in if isCompatible xys' then [fromPairs' xys'] else []
else let (x:xs):p1'' = p1'
ys:p2'' = p2'
in concat [dfs ((x,y):xys)
(refine' (xs : p1'') (dps M.! x))
(refine' ((L.delete y ys):p2'') (dps M.! y))
| y <- ys]
isCompatible xys = and [([x,x'] `S.member` es') == (L.sort [y,y'] `S.member` es') | (x,y) <- xys, (x',y') <- xys, x < x']
dps = M.fromList [(v, distancePartition g v) | v <- vs]
es' = S.fromList es

-- contrary to first thought, you can't stop when a level is null - eg kb 2 3, the third level is null, but the fourth isn't

-- an example for equitable partitions
-- this is a graph whose distance partition (from any vertex) can be refined to an equitable partition
eqgraph = G vs es where
vs = [1..14]
es = L.sort \$ [[1,14],[2,13]] ++ [ [v1,v2] | [v1,v2] <- combinationsOf 2 vs, v1+1 == v2 || v1+3 == v2 && even v2]

-- refine a partition to give an equitable partition
toEquitable g cells = L.sort \$ toEquitable' [] cells where
toEquitable' ls (r:rs) =
let (lls,lrs) = L.partition isSingleton \$ map (splitNumNbrs r) ls
-- so the lrs split, and the lls didn't
rs' = concatMap (splitNumNbrs r) rs
in if isSingleton r -- then we know it won't split further, so can remove it from further processing
then r : toEquitable' (concat lls) (concat lrs ++ rs')
else toEquitable' (r : concat lls) (concat lrs ++ rs')
toEquitable' ls [] = ls
splitNumNbrs t c = map (map snd) \$ L.groupBy (\x y -> fst x == fst y) \$ L.sort
[ (length ((nbrs_g M.! v) `intersect` t), v) | v <- c]
nbrs_g = M.fromList [(v, nbrs g v) | v <- vertices g]

-- try to refine two partitions in parallel, failing if they become mismatched
toEquitable2 nbrs_g psrc ptrg = unzip \$ L.sort \$ toEquitable' [] (zip psrc ptrg) where
toEquitable' ls (r:rs) =
let ls' = map (splitNumNbrs nbrs_g r) ls
(lls,lrs) = L.partition isSingleton \$ map fromJust ls'
rs' = map (splitNumNbrs nbrs_g r) rs
in if any isNothing ls' || any isNothing rs'
then []
else
{- if (isSingleton . fst) r
then r : toEquitable' (concat lls) (concat lrs ++ concatMap fromJust rs')
else -} toEquitable' (r : concat lls) (concat lrs ++ concatMap fromJust rs')
toEquitable' ls [] = ls

splitNumNbrs nbrs_g (t_src,t_trg) (c_src,c_trg) =
let src_split = L.groupBy (\x y -> fst x == fst y) \$ L.sort
[ (length ((nbrs_g M.! v) `intersect` t_src), v) | v <- c_src]
trg_split = L.groupBy (\x y -> fst x == fst y) \$ L.sort
[ (length ((nbrs_g M.! v) `intersect` t_trg), v) | v <- c_trg]
in if map length src_split == map length trg_split
&& map (fst . head) src_split == map (fst . head) trg_split
then Just \$ zip (map (map snd) src_split) (map (map snd) trg_split)
else Nothing
-- else error (show (src_split, trg_split)) -- for debugging

-- Now, every time we intersect two partitions, refine to an equitable partition
-- |Given a graph g, @graphAuts g@ returns a strong generating set for the automorphism group of g.
--
-- Note that the implementation is currently only valid for connected graphs
graphAuts :: (Ord a) => Graph a -> [Permutation a]
graphAuts g@(G vs es)
| isConnected g = graphAuts' [] (toEquitable g \$ valencyPartition g)
| otherwise = error "graphAuts: only implemented for connected graphs"
where graphAuts' us p@((x:ys):pt) =
let p' = L.sort \$ filter (not . null) \$ refine' (ys:pt) (dps M.! x)
in level us p x ys []
++ graphAuts' (x:us) p'
graphAuts' us ([]:pt) = graphAuts' us pt
graphAuts' _ [] = []
level us p@(ph:pt) x (y:ys) hs =
let px = refine' (L.delete x ph : pt) (dps M.! x)
py = refine' (L.delete y ph : pt) (dps M.! y)
uus = zip us us
in case dfsEquitable (dps,es',nbrs_g) ((x,y):uus) px py of
[]  -> level us p x ys hs
h:_ -> let hs' = h:hs in h : level us p x (ys L.\\ (x .^^ hs')) hs'
level _ _ _ [] _ = []
dps = M.fromList [(v, distancePartition g v) | v <- vs]
es' = S.fromList es
nbrs_g = M.fromList [(v, nbrs g v) | v <- vs]
-- To handle disconnected graphs, you not only need to find auts of each component,
-- you also need to find auts that swap components

dfsEquitable (dps,es',nbrs_g) xys p1 p2 = dfs xys p1 p2 where
dfs xys p1 p2
| map length p1 /= map length p2 = []
| otherwise =
let p1' = filter (not . null) p1
p2' = filter (not . null) p2
(p1e,p2e) = toEquitable2 nbrs_g p1' p2'
in if null p1e
then []
else
if all isSingleton p1e
then let xys' = xys ++ zip (concat p1e) (concat p2e)
in if isCompatible xys' then [fromPairs' xys'] else []
else let (x:xs):p1'' = p1e
ys:p2'' = p2e
in concat [dfs ((x,y):xys)
(refine' (xs : p1'') (dps M.! x))
(refine' ((L.delete y ys):p2'') (dps M.! y))
| y <- ys]
isCompatible xys = and [([x,x'] `S.member` es') == (L.sort [y,y'] `S.member` es') | (x,y) <- xys, (x',y') <- xys, x < x']

-- AUTS OF INCIDENCE STRUCTURE VIA INCIDENCE GRAPH

-- based on graphAuts as applied to the incidence graph, but modified to avoid point-block crossover auts
-- |Given the incidence graph of an incidence structure between points and blocks
-- (for example, a set system),
-- @incidenceAuts g@ returns a strong generating set for the automorphism group of the incidence structure.
-- The generators are represented as permutations of the points.
-- The incidence graph should be represented with the points on the left and the blocks on the right.
--
-- Note that the implementation is currently only valid for connected incidence graphs
incidenceAuts :: (Ord p, Ord b) => Graph (Either p b) -> [Permutation p]
incidenceAuts g@(G vs es)
| isConnected g = map points (incidenceAuts' [] [vs])
| otherwise = error "incidenceAuts: only implemented for connected incidence graphs"
where points h = fromPairs [(x,y) | (Left x, Left y) <- toPairs h] -- filtering out the action on blocks
incidenceAuts' us p@((x@(Left _):ys):pt) =
-- let p' = L.sort \$ filter (not . null) \$ refine' (ys:pt) (dps M.! x)
let p' = L.sort \$ refine (ys:pt) (dps M.! x)
in level us p x ys []
++ incidenceAuts' (x:us) p'
incidenceAuts' us ([]:pt) = incidenceAuts' us pt
incidenceAuts' _ (((Right _):_):_) = [] -- if we fix all the points, then the blocks must be fixed too
-- incidenceAuts' _ [] = []
level us p@(ph:pt) x (y@(Left _):ys) hs =
let px = refine' (L.delete x ph : pt) (dps M.! x)
py = refine' (L.delete y ph : pt) (dps M.! y)
uus = zip us us
in case dfsEquitable (dps,es',nbrs_g) ((x,y):uus) px py of
[]  -> level us p x ys hs
h:_ -> let hs' = h:hs in h : level us p x (ys L.\\ (x .^^ hs')) hs'
level _ _ _ _ _ = [] -- includes the case where y matches Right _, which can only occur on first level, before we've distance partitioned
dps = M.fromList [(v, distancePartition g v) | v <- vs]
es' = S.fromList es
nbrs_g = M.fromList [(v, nbrs g v) | v <- vs]

-- GRAPH ISOMORPHISMS

-- !! not yet using equitable partitions, so could probably be more efficient

-- graphIsos :: (Ord a, Ord b) => Graph a -> Graph b -> [[(a,b)]]
graphIsos g1 g2
| isConnected g1 && isConnected g2
= concat [dfs [] (distancePartition g1 v1) (distancePartition g2 v2) | v2 <- vertices g2]
| otherwise = error "graphIsos: only implemented for connected graphs"
where v1 = head \$ vertices g1
dfs xys p1 p2
| map length p1 /= map length p2 = []
| otherwise =
let p1' = filter (not . null) p1
p2' = filter (not . null) p2
in if all isSingleton p1'
then let xys' = xys ++ zip (concat p1') (concat p2')
in if isCompatible xys' then [xys'] else []
else let (x:xs):p1'' = p1'
ys:p2'' = p2'
in concat [dfs ((x,y):xys)
(refine' (xs : p1'') (dps1 M.! x))
(refine' ((L.delete y ys):p2'') (dps2 M.! y))
| y <- ys]
isCompatible xys = and [([x,x'] `S.member` es1) == (L.sort [y,y'] `S.member` es2) | (x,y) <- xys, (x',y') <- xys, x < x']
dps1 = M.fromList [(v, distancePartition g1 v) | v <- vertices g1]
dps2 = M.fromList [(v, distancePartition g2 v) | v <- vertices g2]
es1 = S.fromList \$ edges g1
es2 = S.fromList \$ edges g2

-- !! If we're only interested in seeing whether or not two graphs are iso,
-- !! then the cost of calculating distancePartitions may not be warranted
-- !! (see Math.Combinatorics.Poset: orderIsos01 versus orderIsos)
isGraphIso g1 g2 = (not . null) (graphIsos g1 g2)

-- !! deprecate
isIso g1 g2 = (not . null) (graphIsos g1 g2)

-- the following differs from graphIsos in only two ways
-- we avoid Left, Right crossover isos, by insisting that a Left is taken to a Left (first two lines)
-- we return only the action on the Lefts, and unLeft it
-- incidenceIsos :: (Ord p1, Ord b1, Ord p2, Ord b2) =>
--     Graph (Either p1 b1) -> Graph (Either p2 b2) -> [[(p1,p2)]]
incidenceIsos g1 g2
| isConnected g1 && isConnected g2
= concat [dfs [] (distancePartition g1 v1) (distancePartition g2 v2) | v2@(Left _) <- vertices g2]
| otherwise = error "incidenceIsos: only implemented for connected graphs"
where v1@(Left _) = head \$ vertices g1
dfs xys p1 p2
| map length p1 /= map length p2 = []
| otherwise =
let p1' = filter (not . null) p1
p2' = filter (not . null) p2
in if all isSingleton p1'
then let xys' = xys ++ zip (concat p1') (concat p2')
in if isCompatible xys' then [[(x,y) | (Left x, Left y) <- xys']] else []
else let (x:xs):p1'' = p1'
ys:p2'' = p2'
in concat [dfs ((x,y):xys)
(refine' (xs : p1'') (dps1 M.! x))
(refine' ((L.delete y ys):p2'') (dps2 M.! y))
| y <- ys]
isCompatible xys = and [([x,x'] `S.member` es1) == (L.sort [y,y'] `S.member` es2) | (x,y) <- xys, (x',y') <- xys, x < x']
dps1 = M.fromList [(v, distancePartition g1 v) | v <- vertices g1]
dps2 = M.fromList [(v, distancePartition g2 v) | v <- vertices g2]
es1 = S.fromList \$ edges g1
es2 = S.fromList \$ edges g2

isIncidenceIso g1 g2 = (not . null) (incidenceIsos g1 g2)

{-
removeGens x gs = removeGens' [] gs where
baseOrbit = x .^^ gs
removeGens' ls (r:rs) =
if x .^^ (ls++rs) == baseOrbit
then removeGens' ls rs
else removeGens' (r:ls) rs
removeGens' ls [] = reverse ls
-- !! reverse is probably pointless

-- !! DON'T THINK THIS IS WORKING PROPERLY
-- eg graphAutsSGSNew \$ toGraph ([1..7],[[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])
-- returns [[[1,2]],[[5,6]],[[5,7,6]],[[6,7]]]
-- whereas [[6,7]] was a Schreier generator, so shouldn't have been listed

-- Using Schreier generators to seed the next level
-- At the moment this is slower than the above
-- (This could be modified to allow us to start the search with a known subgroup)
graphAutsNew g@(G vs es) = graphAuts' [] [] [vs] where
graphAuts' us hs p@((x:ys):pt) =
let ys' = ys L.\\ (x .^^ hs) -- don't need to consider points which can already be reached from Schreier generators
hs' = level us p x ys' []
p' = L.sort \$ filter (not . null) \$ refine' (ys:pt) (dps M.! x)
reps = cosetRepsGx (hs'++hs) x
schreierGens = removeGens x \$ schreierGeneratorsGx (x,reps) (hs'++hs)
in hs' ++ graphAuts' (x:us) schreierGens p'
graphAuts' us hs ([]:pt) = graphAuts' us hs pt
graphAuts' _ _ [] = []
level us p@(ph:pt) x (y:ys) hs =
let px = refine' (L.delete x ph : pt) (dps M.! x)
py = refine' (L.delete y ph : pt) (dps M.! y)
uus = zip us us
in if map length px /= map length py
then level us p x ys hs
else case dfs ((x,y):uus) (filter (not . null) px) (filter (not . null) py) of
[]  -> level us p x ys hs
h:_ -> let hs' = h:hs in h : level us p x (ys L.\\ (x .^^ hs')) hs'
-- if h1 = (1 2)(3 4), and h2 = (1 3 2), then we can remove 4 too
level _ _ _ [] _ = []
dfs xys p1 p2
| map length p1 /= map length p2 = []
| otherwise =
let p1' = filter (not . null) p1
p2' = filter (not . null) p2
in if all isSingleton p1'
then let xys' = xys ++ zip (concat p1') (concat p2')
in if isCompatible xys' then [fromPairs' xys'] else []
else let (x:xs):p1'' = p1'
ys:p2'' = p2'
in concat [dfs ((x,y):xys)
(refine' (xs : p1'') (dps M.! x))
(refine' ((L.delete y ys):p2'') (dps M.! y))
| y <- ys]
isCompatible xys = and [([x,x'] `S.member` es') == (L.sort [y,y'] `S.member` es') | (x,y) <- xys, (x',y') <- xys, x < x']
dps = M.fromList [(v, distancePartition g v) | v <- vs]
es' = S.fromList es
-}

```