-- Copyright (c) 2008-2011, David Amos. All rights reserved. -- |A module defining a polymorphic data type for (simple, undirected) graphs, -- together with constructions of some common families of graphs, -- new from old constructions, and calculation of simple properties of graphs. module Math.Combinatorics.Graph where import qualified Data.List as L import Data.Maybe (isJust) import qualified Data.Map as M import qualified Data.Set as S import Control.Arrow ( (&&&) ) import Math.Common.ListSet as LS import Math.Core.Utils import Math.Algebra.Group.PermutationGroup hiding (fromDigits, fromBinary) import qualified Math.Algebra.Group.SchreierSims as SS -- Main source: Godsil & Royle, Algebraic Graph Theory -- COMBINATORICS -- Some functions we'll use set xs = map head $ L.group $ L.sort xs -- subsets of a set (returned in "binary" order) powerset [] = [[]] powerset (x:xs) = let p = powerset xs in p ++ map (x:) p -- GRAPH -- |Datatype for graphs, represented as a list of vertices and a list of edges. -- For most purposes, graphs are required to be in normal form. -- A graph G vs es is in normal form if (i) vs is in ascending order without duplicates, -- (ii) es is in ascending order without duplicates, (iii) each e in es is a 2-element list [x,y], x<y data Graph a = G [a] [[a]] deriving (Eq,Ord,Show) instance Functor Graph where -- |If f is not order-preserving, then you should call nf afterwards fmap f (G vs es) = G (map f vs) (map (map f) es) -- could use DeriveFunctor to derive this Functor instance -- |Convert a graph to normal form. The input is assumed to be a valid graph apart from order nf :: Ord a => Graph a -> Graph a nf (G vs es) = G vs' es' where vs' = L.sort vs es' = L.sort (map L.sort es) -- we require that vs, es, and each individual e are sorted isSetSystem xs bs = isListSet xs && isListSet bs && all isListSet bs && all (`isSubset` xs) bs isGraph vs es = isSetSystem vs es && all ( (==2) . length) es -- |Safe constructor for graph from lists of vertices and edges. -- graph (vs,es) checks that vs and es are valid before returning the graph. graph :: (Ord t) => ([t], [[t]]) -> Graph t graph (vs,es) | isGraph vs es = G vs es -- | otherwise = error ( "graph " ++ show (vs,es) ) -- isValid g = g where g = G vs es toGraph (vs,es) | isGraph vs' es' = G vs' es' where vs' = L.sort vs es' = L.sort $ map L.sort es -- note that calling isListSet on a sorted list still checks that there are no duplicates vertices (G vs _) = vs edges (G _ es) = es -- OTHER REPRESENTATIONS -- incidence matrix of a graph -- (rows and columns indexed by edges and vertices respectively) -- (warning: in the literature it is often the other way round) incidenceMatrix (G vs es) = [ [if v `elem` e then 1 else 0 | v <- vs] | e <- es] fromIncidenceMatrix m = graph (vs,es) where n = L.genericLength $ head m vs = [1..n] es = L.sort $ map edge m edge row = [v | (1,v) <- zip row vs] adjacencyMatrix (G vs es) = [ [if L.sort [i,j] `S.member` es' then 1 else 0 | j <- vs] | i <- vs] where es' = S.fromList es fromAdjacencyMatrix m = graph (vs,es) where n = L.genericLength m vs = [1..n] es = es' 1 m es' i (r:rs) = [ [i,j] | (j,1) <- drop i (zip vs r)] ++ es' (i+1) rs es' _ [] = [] -- SOME SIMPLE FAMILIES OF GRAPHS -- |The null graph on n vertices is the graph with no edges nullGraph :: (Integral t) => t -> Graph t nullGraph n = G [1..n] [] -- |The null graph, with no vertices or edges nullGraph' :: Graph Int -- type signature needed nullGraph' = G [] [] -- |c n is the cyclic graph on n vertices c :: (Integral t) => t -> Graph t c n | n >= 3 = graph (vs,es) where vs = [1..n] es = L.insert [1,n] [[i,i+1] | i <- [1..n-1]] -- automorphism group is D2n -- |k n is the complete graph on n vertices k :: (Integral t) => t -> Graph t k n = graph (vs,es) where vs = [1..n] es = [[i,j] | i <- [1..n-1], j <- [i+1..n]] -- == combinationsOf 2 [1..n] -- automorphism group is Sn -- |kb m n is the complete bipartite graph on m and n vertices kb :: (Integral t) => t -> t -> Graph t kb m n = to1n $ kb' m n -- |kb' m n is the complete bipartite graph on m left and n right vertices kb' :: (Integral t) => t -> t -> Graph (Either t t) kb' m n = graph (vs,es) where vs = map Left [1..m] ++ map Right [1..n] es = [ [Left i, Right j] | i <- [1..m], j <- [1..n] ] -- automorphism group is Sm*Sn (plus a flip if m==n) -- |q k is the graph of the k-cube q :: (Integral t) => Int -> Graph t q k = fromBinary $ q' k q' :: (Integral t) => Int -> Graph [t] q' k = graph (vs,es) where vs = sequence $ replicate k [0,1] -- ptsAn k f2 es = [ [u,v] | [u,v] <- combinationsOf 2 vs, hammingDistance u v == 1 ] hammingDistance as bs = length $ filter id $ zipWith (/=) as bs -- can probably type-coerce this to be Graph [F2] if required tetrahedron = k 4 cube = q 3 octahedron = graph (vs,es) where vs = [1..6] es = combinationsOf 2 vs L.\\ [[1,6],[2,5],[3,4]] dodecahedron = toGraph (vs,es) where vs = [1..20] es = [ [1,2],[2,3],[3,4],[4,5],[5,1], [6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,6], [16,17],[17,18],[18,19],[19,20],[20,16], [1,6],[2,8],[3,10],[4,12],[5,14], [7,16],[9,17],[11,18],[13,19],[15,20] ] icosahedron = toGraph (vs,es) where vs = [1..12] es = [ [1,2],[1,3],[1,4],[1,5],[1,6], [2,3],[3,4],[4,5],[5,6],[6,2], [7,12],[8,12],[9,12],[10,12],[11,12], [7,8],[8,9],[9,10],[10,11],[11,7], [2,7],[7,3],[3,8],[8,4],[4,9],[9,5],[5,10],[10,6],[6,11],[11,2] ] -- Prisms are regular, vertex-transitive, but not edge-transitive unless n == 1, 2, 4. -- (prism 2 ~= q 2, prism 4 ~= q 3) prism :: Int -> Graph (Int,Int) prism n = k 2 `cartProd` c n -- convert a graph to have [1..n] as vertices to1n (G vs es) = graph (vs',es') where mapping = M.fromList $ zip vs [1..] -- the mapping from vs to [1..n] vs' = M.elems mapping es' = [map (mapping M.!) e | e <- es] -- the edges will already be sorted correctly by construction -- |Given a graph with vertices which are lists of small integers, eg [1,2,3], -- return a graph with vertices which are the numbers obtained by interpreting these as digits, eg 123. -- The caller is responsible for ensuring that this makes sense (eg that the small integers are all < 10) fromDigits :: Integral a => Graph [a] -> Graph a fromDigits = fmap fromDigits' {- fromDigits (G vs es) = graph (vs',es') where vs' = map fromDigits' vs es' = (map . map) fromDigits' es -} -- |Given a graph with vertices which are lists of 0s and 1s, -- return a graph with vertices which are the numbers obtained by interpreting these as binary digits. -- For example, [1,1,0] -> 6. fromBinary :: Integral a => Graph [a] -> Graph a fromBinary = fmap fromBinary' {- fromBinary (G vs es) = graph (vs',es') where vs' = map fromBinary' vs es' = (map . map) fromBinary' es -} petersen :: Graph [Integer] petersen = graph (vs,es) where vs = combinationsOf 2 [1..5] es = [ [v1,v2] | [v1,v2] <- combinationsOf 2 vs, disjoint v1 v2] -- == kneser 5 2 == j 5 2 0 -- == complement $ lineGraph' $ k 5 -- == complement $ t' 5 -- NEW GRAPHS FROM OLD complement :: (Ord t) => Graph t -> Graph t complement (G vs es) = graph (vs,es') where es' = combinationsOf 2 vs LS.\\ es -- es' = [e | e <- combinationsOf 2 vs, e `notElem` es] -- |The restriction of a graph to a subset of the vertices restriction :: (Eq a) => Graph a -> [a] -> Graph a restriction g@(G vs es) us = G us (es `restrict` us) where es `restrict` us = [e | e@[i,j] <- es, i `elem` us, j `elem` us] inducedSubgraph :: (Eq a) => Graph a -> [a] -> Graph a inducedSubgraph g@(G vs es) us = G us (es `restrict` us) where es `restrict` us = [e | e@[i,j] <- es, i `elem` us, j `elem` us] lineGraph g = to1n $ lineGraph' g lineGraph' (G vs es) = graph (es, [ [ei,ej] | ei <- es, ej <- dropWhile (<= ei) es, ei `intersect` ej /= [] ]) -- For example cartProd (c m) (c n) is a wireframe for a torus -- cartProd (q m) (q n) `isGraphIso` q (m+n) -- Godsil and Royle p154 cartProd (G vs es) (G vs' es') = G us [e | e@[u,u'] <- combinationsOf 2 us, u `adj` u' ] where us = [(v,v') | v <- vs, v' <- vs'] eset = S.fromList es eset' = S.fromList es' adj (x1,y1) (x2,y2) = x1 == x2 && L.sort [y1,y2] `S.member` eset' || y1 == y2 && L.sort [x1,x2] `S.member` eset -- SIMPLE PROPERTIES OF GRAPHS order = length . vertices size = length . edges -- also called degree valency (G vs es) v = length $ filter (v `elem`) es valencies g@(G vs es) = map (head &&& length) $ L.group $ L.sort $ map (valency g) vs valencyPartition g@(G vs es) = map (map snd) $ L.groupBy (\x y -> fst x == fst y) [(valency g v, v) | v <- vs] regularParam g = case valencies g of [(v,_)] -> Just v _ -> Nothing -- |A graph is regular if all vertices have the same valency (degree) isRegular :: (Eq t) => Graph t -> Bool isRegular g = isJust $ regularParam g -- |A 3-regular graph is called a cubic graph isCubic :: (Eq t) => Graph t -> Bool isCubic g = regularParam g == Just 3 nbrs (G vs es) v = [u | [u,v'] <- es, v == v'] ++ [w | [v',w] <- es, v == v'] -- if the graph is valid, then the neighbours will be returned in ascending order -- find paths from x to y using bfs -- by definition, a path is a subgraph isomorphic to a "line" - it can't have self-crossings -- (a walk allows self-crossings, a trail allows self-crossings but no edge reuse) findPaths g@(G vs es) x y = map reverse $ bfs [ [x] ] where bfs ((z:zs) : nodes) | z == y = (z:zs) : bfs nodes | otherwise = bfs (nodes ++ [(w:z:zs) | w <- nbrs g z, w `notElem` zs]) bfs [] = [] -- |Within a graph G, the distance d(u,v) between vertices u, v is length of the shortest path from u to v distance :: (Eq a) => Graph a -> a -> a -> Int distance g x y = case findPaths g x y of [] -> -1 -- infinite p:ps -> length p - 1 -- |The diameter of a graph is maximum distance between two distinct vertices diameter :: (Ord t) => Graph t -> Int diameter g@(G vs es) | isConnected g = maximum $ map maxDistance vs | otherwise = -1 where maxDistance v = length (distancePartition g v) - 1 -- find cycles starting at x -- by definition, a cycle is a subgraph isomorphic to a cyclic graph - it can't have self-crossings -- (a circuit allows self-crossings but not edge reuse) findCycles g@(G vs es) x = [reverse (x:z:zs) | z:zs <- bfs [ [x] ], z `elem` nbrsx, length zs > 1] where nbrsx = nbrs g x bfs ((z:zs) : nodes) = (z:zs) : bfs (nodes ++ [ w:z:zs | w <- nbrs g z, w `notElem` zs]) bfs [] = [] -- |The girth of a graph is the size of the smallest cycle that it contains. -- Note: If the graph contains no cycles, we return -1, representing infinity. girth :: (Eq t) => Graph t -> Int girth g@(G vs es) = minimum' $ map minCycle vs where minimum' xs = let (zs,nzs) = L.partition (==0) xs in if null nzs then -1 else minimum nzs minCycle v = case findCycles g v of [] -> 0 c:cs -> length c - 1 -- because v occurs twice in c, as startpoint and endpoint -- circumference = max cycle - Bollobas p104 -- Vertices that are not in the same component as the start vertex all go into a final cell distancePartition g@(G vs es) v = distancePartitionS vs (S.fromList es) v distancePartitionS vs eset v = distancePartition' (S.singleton v) (S.delete v (S.fromList vs)) where distancePartition' boundary exterior | S.null boundary = if S.null exterior then [] else [S.toList exterior] -- graph may not have been connected | otherwise = let (boundary', exterior') = S.partition (\v -> any (`S.member` eset) [L.sort [u,v] | u <- S.toList boundary]) exterior in S.toList boundary : distancePartition' boundary' exterior' -- the connected component to which v belongs component g v = component' S.empty (S.singleton v) where component' interior boundary | S.null boundary = S.toList interior | otherwise = let interior' = S.union interior boundary boundary' = foldl S.union S.empty [S.fromList (nbrs g x) | x <- S.toList boundary] S.\\ interior' in component' interior' boundary' -- TODO: This can almost certainly be made more efficient. -- nbrs is O(n), and this calls it for each vertex in the component, so it is O(n^2) -- |Is the graph connected? isConnected :: (Ord t) => Graph t -> Bool isConnected g@(G (v:vs) es) = length (component g v) == length (v:vs) isConnected (G [] []) = True components g = components' (vertices g) where components' [] = [] components' (v:vs) = let c = component g v in c : components' (vs LS.\\ c) -- MORE GRAPHS -- Generalized Johnson graph, Godsil & Royle p9 -- Also called generalised Kneser graph, http://en.wikipedia.org/wiki/Kneser_graph j v k i | v >= k && k >= i = graph (vs,es) where vs = combinationsOf k [1..v] es = [ [v1,v2] | [v1,v2] <- combinationsOf 2 vs, length (v1 `intersect` v2) == i ] -- j v k i is isomorphic to j v (v-k) (v-2k+i), so may as well have v >= 2k -- kneser v k | v >= 2*k = j v k 0 -- |kneser n k returns the kneser graph KG n,k - -- whose vertices are the k-element subsets of [1..n], with edges between disjoint subsets kneser :: Int -> Int -> Graph [Int] kneser n k | 2*k <= n = graph (vs,es) where vs = combinationsOf k [1..n] es = [ [v1,v2] | [v1,v2] <- combinationsOf 2 vs, disjoint v1 v2] johnson v k | v >= 2*k = j v k (k-1) bipartiteKneser n k | 2*k < n = graph (vs,es) where vs = map Left (combinationsOf k [1..n]) ++ map Right (combinationsOf (n-k) [1..n]) es = [ [Left u, Right v] | u <- combinationsOf k [1..n], v <- combinationsOf (n-k) [1..n], u `isSubset` v] desargues1 = bipartiteKneser 5 2 -- Generalised Petersen graphs -- http://en.wikipedia.org/wiki/Petersen_graph gp n k | 2*k < n = toGraph (vs,es) where vs = map Left [0..n-1] ++ map Right [0..n-1] es = (map . map) Left [ [i, (i+1) `mod` n] | i <- [0..n-1] ] ++ [ [Left i, Right i] | i <- [0..n-1] ] ++ (map . map) Right [ [i, (i+k) `mod` n] | i <- [0..n-1] ] petersen2 = gp 5 2 prism' n = gp n 1 durer = gp 6 2 mobiusKantor = gp 8 3 dodecahedron2 = gp 10 2 desargues2 = gp 10 3