-- Copyright (c) 2009, David Amos. All rights reserved. -- |A module defining a type for hypergraphs. module Math.Combinatorics.Hypergraph where import qualified Data.List as L import Math.Common.ListSet import Math.Combinatorics.Graph hiding (incidenceMatrix) import Math.Algebra.Group.PermutationGroup (orbitB, p) -- needed for construction of Coxeter group -- not used in this module, only in GHCi import Math.Algebra.Field.Base import Math.Algebra.Field.Extension import Math.Combinatorics.Design hiding (incidenceMatrix, incidenceGraph, dual, isSubset, fanoPlane) -- set system or hypergraph data Hypergraph a = H [a] [[a]] deriving (Eq,Ord,Show) hypergraph xs bs | isSetSystem xs bs = H xs bs toHypergraph xs bs = H xs' bs' where xs' = L.sort xs bs' = L.sort $ map L.sort bs -- this still doesn't guarantee that all bs are subset of xs -- |Is this hypergraph uniform - meaning that all blocks are of the same size isUniform :: (Ord a) => Hypergraph a -> Bool isUniform h@(H xs bs) = isSetSystem xs bs && same (map length bs) same (x:xs) = all (==x) xs same [] = True fromGraph (G vs es) = H vs es fromDesign (D xs bs) = H xs (L.sort bs) -- !! should insist that designs have blocks in order -- !! dual probably doesn't guarantee this at present {- dual (H xs bs) = toHypergraph (bs, map beta xs) where beta x = filter (x `elem`) bs -} -- INCIDENCE GRAPH -- data Incidence a = P a | B [a] deriving (Eq, Ord, Show) -- compare Design, where we just use Left, Right -- Also called the Levi graph incidenceGraph :: (Ord a) => Hypergraph a -> Graph (Either a [a]) incidenceGraph (H xs bs) = G vs es where vs = map Left xs ++ map Right bs es = L.sort [ [Left x, Right b] | b <- bs, x <- b] -- INCIDENCE MATRIX -- !! why are we doing this the other way round to the literature ?? -- incidence matrix of a hypergraph -- (rows and columns indexed by edges and vertices respectively) -- (warning: in the literature it is often the other way round) incidenceMatrix (H vs es) = [ [if v `elem` e then 1 else 0 | v <- vs] | e <- es] fromIncidenceMatrix m = H 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] -- isTwoGraph -- We can represent various incidence structures as hypergraphs, -- by identifying the lines with the sets of points that they contain isPartialLinearSpace :: (Ord a) => Hypergraph a -> Bool isPartialLinearSpace h@(H ps ls) = isSetSystem ps ls && all ( (<=1) . length ) [filter (pair `isSubset`) ls | pair <- combinationsOf 2 ps] -- any two points are incident with at most one line -- Godsil & Royle, p79 -- |Is this hypergraph a projective plane - meaning that any two lines meet in a unique point, -- and any two points lie on a unique line isProjectivePlane :: (Ord a) => Hypergraph a -> Bool isProjectivePlane h@(H ps ls) = isSetSystem ps ls && all ( (==1) . length) [intersect l1 l2 | [l1,l2] <- combinationsOf 2 ls] && -- any two lines meet in a unique point all ( (==1) . length) [ filter ([p1,p2] `isSubset`) ls | [p1,p2] <- combinationsOf 2 ps] -- any two points lie in a unique line -- |Is this hypergraph a projective plane with a triangle. -- This is a weak non-degeneracy condition, which eliminates all points on the same line, or all lines through the same point. isProjectivePlaneTri :: (Ord a) => Hypergraph a -> Bool isProjectivePlaneTri h@(H ps ls) = isProjectivePlane h && any triangle (combinationsOf 3 ps) where triangle t@[p1,p2,p3] = (not . null) [l | l <- ls, [p1,p2] `isSubset` l, p3 `notElem` l] && -- there is a line containing p1,p2 but not p3 (not . null) [l | l <- ls, [p1,p3] `isSubset` l, p2 `notElem` l] && (not . null) [l | l <- ls, [p2,p3] `isSubset` l, p1 `notElem` l] -- |Is this hypergraph a projective plane with a quadrangle. -- This is a stronger non-degeneracy condition. isProjectivePlaneQuad :: (Ord a) => Hypergraph a -> Bool isProjectivePlaneQuad h@(H ps ls) = isProjectivePlane h && any quadrangle (combinationsOf 4 ps) where quadrangle q = all (not . collinear) (combinationsOf 3 q) -- no three points collinear collinear ps = any (ps `isSubset`) ls -- > isProjectivePlaneQuad $ fromDesign $ pg2 f2 -- True -- GENERALIZED QUADRANGLES -- Godsil & Royle p81 isGeneralizedQuadrangle :: (Ord a) => Hypergraph a -> Bool isGeneralizedQuadrangle h@(H ps ls) = isPartialLinearSpace h && all (\(l,p) -> unique [p' | p' <- l, collinear (pair p p')]) [(l,p) | l <- ls, p <- ps, p `notElem` l] && -- given any line l and point p not on l, there is a unique point p' on l with p and p' collinear any (not . collinear) (powerset ps) && -- there are non collinear points any (not . concurrent) (powerset ls) -- there are non concurrent lines where unique xs = length xs == 1 pair x y = if x < y then [x,y] else [y,x] collinear ps = any (ps `isSubset`) ls concurrent ls = any (\p -> all (p `elem`) ls) ps grid m n = H ps ls where ps = [(i,j) | i <- [1..m], j <- [1..n] ] ls = L.sort $ [ [(i,j) | i <- [1..m] ] | j <- [1..n] ] -- horizontal lines ++ [ [(i,j) | j <- [1..n] ] | i <- [1..m] ] -- vertical lines dualGrid m n = fromGraph $ kb m n -- the lines of the grid are the points of the dual, and the points of the grid are the lines of the dual isGenQuadrangle' h = diameter g == 4 && girth g == 8 -- !! plus non-degeneracy conditions where g = incidenceGraph h -- CONFIGURATIONS -- http://en.wikipedia.org/wiki/Projective_configuration -- |Is this hypergraph a (projective) configuration. isConfiguration :: (Ord a) => Hypergraph a -> Bool isConfiguration h@(H ps ls) = isUniform h && -- a set system, with each line incident with the same number of points same [length (filter (p `elem`) ls) | p <- ps] -- each point is incident with the same number of lines fanoPlane :: Hypergraph Integer fanoPlane = toHypergraph [1..7] [[1,2,4],[2,3,5],[3,4,6],[4,5,7],[5,6,1],[6,7,2],[7,1,3]] -- |The Heawood graph is the incidence graph of the Fano plane heawoodGraph :: Graph (Either Integer [Integer]) heawoodGraph = incidenceGraph fanoPlane desarguesConfiguration :: Hypergraph [Integer] desarguesConfiguration = H xs bs where xs = combinationsOf 2 [1..5] bs = [ [x | x <- xs, x `isSubset` b] | b <- combinationsOf 3 [1..5] ] desarguesGraph :: Graph (Either [Integer] [[Integer]]) desarguesGraph = incidenceGraph desarguesConfiguration pappusConfiguration :: Hypergraph Integer pappusConfiguration = H xs bs where xs = [1..9] bs = L.sort [ [1,2,3], [4,5,6], [7,8,9], [1,5,9], [1,6,8], [2,4,9], [3,4,8], [2,6,7], [3,5,7] ] pappusGraph :: Graph (Either Integer [Integer]) pappusGraph = incidenceGraph pappusConfiguration -- !! no particular reason why the following is here rather than elsewhere {- triples = combinationsOf 3 [1..7] heptads = [ [a,b,c,d,e,f,g] | a <- triples, b <- triples, a < b, meetOne b a, c <- triples, b < c, all (meetOne c) [a,b], d <- triples, c < d, all (meetOne d) [a,b,c], e <- triples, d < e, all (meetOne e) [a,b,c,d], f <- triples, e < f, all (meetOne f) [a,b,c,d,e], g <- triples, f < g, all (meetOne g) [a,b,c,d,e,f], foldl intersect [1..7] [a,b,c,d,e,f,g] == [] ] where meetOne x y = length (intersect x y) == 1 -- each pair of triples meet in exactly one point, and there is no point in all of them - Godsil & Royle p69 -- (so these are the projective planes over 7 points) -} -- Godsil & Royle p69 coxeterGraph :: Graph [Integer] coxeterGraph = G vs es where g = p [[1..7]] vs = L.sort $ concatMap (orbitB [g]) [[1,2,4],[3,5,7],[3,6,7],[5,6,7]] es = [ e | e@[v1,v2] <- combinationsOf 2 vs, disjoint v1 v2] -- is this the incidence graph of a hypergraph involving heptads over triples? -- edges of K6 duads = combinationsOf 2 [1..6] -- 1-factors of K6 -- 15 different ways to pick three disjoint duads from [1..6] synthemes = [ [d1,d2,d3] | d1 <- duads, d2 <- duads, d2 > d1, disjoint d1 d2, d3 <- duads, d3 > d2, disjoint d1 d3, disjoint d2 d3 ] -- |The Tutte-Coxeter graph, also called the Tutte 8-cage tutteCoxeterGraph :: Graph (Either [Integer] [[Integer]]) tutteCoxeterGraph = incidenceGraph $ H duads synthemes -- Also known as line graph intersectionGraph (H xs bs) = G vs es where vs = bs es = [pair | pair@[b1,b2] <- combinationsOf 2 bs, not (disjoint b1 b2)]