-- | For Connectivity analysis purposes a 'DGraph' can be converted into a -- | 'UGraph' using 'toUndirected' {-# LANGUAGE ScopedTypeVariables #-} module Data.Graph.Connectivity where import Data.List (foldl') import Data.Hashable import qualified Data.Set as S import Data.Graph.DGraph import Data.Graph.Types import Data.Graph.UGraph -- | Tell if two vertices of a graph are connected -- -- Two vertices are @connected@ if it exists a path between them. The order of -- the vertices is relevant when the graph is directed areConnected :: forall g v e . (Graph g, Hashable v, Eq v, Ord v) => g v e -> v -> v -> Bool areConnected g fromV toV | fromV == toV = True | otherwise = search (fromV : reachableAdjacentVertices g fromV) S.empty toV where search :: [v] -> S.Set v -> v -> Bool search [] _ _ = False search (v:vs) banned v' | v `S.member` banned = search vs banned v' | v == v' = True | otherwise = search (v : reachableAdjacentVertices g v) banned' v' || search vs banned' v' where banned' = v `S.insert` banned -- | Opposite of 'areConnected' areDisconnected :: (Graph g, Hashable v, Eq v, Ord v) => g v e -> v -> v -> Bool areDisconnected g fromV toV = not \$ areConnected g fromV toV -- | Tell if a vertex is isolated -- -- A vertex is @isolated@ if it has no incident edges, that is, it has a degree -- of zero isIsolated :: (Graph g, Hashable v, Eq v) => g v e -> v -> Bool isIsolated g v = vertexDegree g v == 0 -- | Tell if a graph is connected -- -- An undirected graph is @connected@ when there is a path between every pair -- of vertices -- FIXME: Use a O(n) algorithm isConnected :: (Graph g, Hashable v, Eq v, Ord v) => g v e -> Bool isConnected g = go vs True where vs = vertices g go _ False = False go [] bool = bool go (v':vs') bool = go vs' \$ foldl' (\b v -> b && areConnected g v v') bool vs -- | Tell if a graph is bridgeless -- -- A graph is @bridgeless@ if it has no edges that, when removed, split the -- graph in two isolated components -- FIXME: Use a O(n) algorithm isBridgeless :: (Hashable v, Eq v, Ord v) => UGraph v e -> Bool isBridgeless g = foldl' (\b vs -> b && isConnected (removeEdgePair vs g)) True (edgePairs g) -- | Tell if a 'UGraph' is orientable -- -- An undirected graph is @orientable@ if it can be converted into a directed -- graph that is @strongly connected@ (See 'isStronglyConnected') isOrientable :: (Hashable v, Eq v, Ord v) => UGraph v e -> Bool isOrientable g = isConnected g && isBridgeless g -- | Tell if a 'DGraph' is weakly connected -- -- A directed graph is @weakly connected@ if the underlying undirected graph -- is @connected@ isWeaklyConnected :: (Hashable v, Eq v, Ord v) => DGraph v e -> Bool isWeaklyConnected = isConnected . toUndirected -- | Tell if a 'DGraph' is strongly connected -- -- A directed graph is @strongly connected@ if it contains a directed path -- on every pair of vertices isStronglyConnected :: (Hashable v, Eq v, Ord v) => DGraph v e -> Bool isStronglyConnected = isConnected -- TODO -- connected component -- strong components -- vertex cut -- vertex connectivity -- biconnectivity -- triconnectivity -- separable -- bridge -- edge-connectivity -- maximally connected -- maximally edge-connected -- super-connectivity -- hyper-connectivity -- Menger's theorem -- Robin's Theorem: a graph is orientable if it is connected and has no bridges