# 99 questions/Solutions/85

### From HaskellWiki

(**) Graph isomorphism

Two graphs G1(N1,E1) and G2(N2,E2) are isomorphic if there is a bijection f: N1 -> N2 such that for any nodes X,Y of N1, X and Y are adjacent if and only if f(X) and f(Y) are adjacent.

Write a predicate that determines whether two graphs are isomorphic.

This solution compares the canonical forms of the two graphs to determine whether they are isomorphic.

data Graph a = Graph [a] [(a, a)] deriving (Show, Eq) data Adjacency a = Adj [(a, [a])] deriving (Show, Eq) graphG1 = Graph [1, 2, 3, 4, 5, 6, 7, 8] [(1, 5), (1, 6), (1, 7), (2, 5), (2, 6), (2, 8), (3, 5), (3, 7), (3, 8), (4, 6), (4, 7), (4, 8)] graphH1 = Graph [1, 2, 3, 4, 5, 6, 7, 8] [(1, 2), (1, 4), (1, 5), (6, 2), (6, 5), (6, 7), (8, 4), (8, 5), (8, 7), (3, 2), (3, 4), (3, 7)] graphToAdj :: (Eq a) => Graph a -> Adjacency a graphToAdj (Graph [] _) = Adj [] graphToAdj (Graph (x:xs) ys) = Adj ((x, ys >>= f) : zs) where f (a, b) | a == x = [b] | b == x = [a] | otherwise = [] Adj zs = graphToAdj (Graph xs ys) iso :: (Ord a, Enum a, Ord b, Enum b) => Graph a -> Graph b -> Bool iso g@(Graph xs ys) h@(Graph xs' ys') = length xs == length xs' && length ys == length ys' && canon g == canon h canon :: (Ord a, Enum a) => Graph a -> String canon g = minimum $ map f $ perm $ length a where Adj a = graphToAdj g v = map fst a perm n = foldr (\x xs -> [i : s | i <- [1..n], s <- xs, i `notElem` s]) [[]] [1..n] f p = let n = zip v p in show [(snd x, sort id $ map (\x -> snd $ head $ snd $ break ((==) x . fst) n) $ snd $ find a x) | x <- sort snd n] sort f n = foldr (\x xs -> let (lt, gt) = break ((<) (f x) . f) xs in lt ++ [x] ++ gt) [] n find a x = let (xs, ys) = break ((==) (fst x) . fst) a in head ys