diff --git a/graphite.cabal b/graphite.cabal
--- a/graphite.cabal
+++ b/graphite.cabal
@@ -1,5 +1,5 @@
 name:                graphite
-version:             0.9.3.0
+version:             0.9.4.0
 synopsis:            Graphs and networks library
 description:         Represent, analyze and visualize graphs
 homepage:            https://github.com/alx741/graphite#readme
@@ -28,6 +28,7 @@
                      , Data.Graph.Visualize
   build-depends:       base >= 4.7 && < 5
                      , QuickCheck
+                     , semigroups
                      , bytestring
                      , cassava
                      , containers
diff --git a/src/Data/Graph/DGraph.hs b/src/Data/Graph/DGraph.hs
--- a/src/Data/Graph/DGraph.hs
+++ b/src/Data/Graph/DGraph.hs
@@ -4,7 +4,7 @@
 
 module Data.Graph.DGraph where
 
-import Data.List      (foldl')
+import Data.List      (foldl', intersect)
 import Data.Semigroup
 import GHC.Generics   (Generic)
 
@@ -34,11 +34,11 @@
         xs <- readPrec
         return (fromList xs)
 
-instance (Hashable v) => Monoid (DGraph v e) where
+instance (Hashable v, Eq v) => Monoid (DGraph v e) where
     mempty = empty
     mappend = union
 
-instance (Hashable v) => Semigroup (DGraph v e) where
+instance (Hashable v, Eq v) => Semigroup (DGraph v e) where
     (<>) = mappend
 
 instance (Hashable v, Eq v) => Functor (DGraph v) where
@@ -110,9 +110,12 @@
         where go bool v = bool && not (HM.member v $ getLinks v $ unDGraph g)
 
 
-    union = undefined
-    intersection = undefined
-    join = undefined
+    union g1 g2 = insertArcs (toArcsList g1) $ insertVertices (vertices g1) g2
+
+    intersection g1 g2 =
+        insertVertices (isolatedVertices g1 `intersect` isolatedVertices g2) $
+        fromArcsList (toArcsList g1 `intersect` toArcsList g2)
+
 
 
     toList (DGraph _ g) = zip vs $ fmap (\v -> HM.toList $ getLinks v g) vs
diff --git a/src/Data/Graph/Types.hs b/src/Data/Graph/Types.hs
--- a/src/Data/Graph/Types.hs
+++ b/src/Data/Graph/Types.hs
@@ -44,7 +44,7 @@
 
     -- * Operations
 
-    -- | Retrieve the vertices of a graph
+    -- | Retrieve all the vertices of a graph
     vertices :: g v e -> [v]
 
     -- | Retrieve the edges of a graph
@@ -161,6 +161,10 @@
     removeEdgePairAndVertices (v1, v2) g =
         removeVertex v2 $ removeVertex v1 $ removeEdgePair (v1, v2) g
 
+    -- | Retrieve the isolated vertices of a graph, if any
+    isolatedVertices :: (Hashable v, Eq v) => g v e -> [v]
+    isolatedVertices g = filter (\v -> vertexDegree g v == 0) $ vertices g
+
     -- | Tell if a graph is simple
     -- | A graph is @simple@ if it has no loops
     isSimple :: (Hashable v, Eq v) => g v e -> Bool
@@ -169,16 +173,10 @@
     -- * Binary operations
 
     -- | Union of two graphs
-    union :: g v e -> g v e -> g v e
+    union :: (Hashable v, Eq v) => g v e -> g v e -> g v e
 
     -- | Intersection of two graphs
-    intersection :: g v e -> g v e -> g v e
-
-    -- | Join two graphs
-    -- |
-    -- | The @join@ of two graphs G1 and G2 is a new graph where each vertex of
-    -- | G1 has an edge to all the vertices of G2
-    join :: g v e -> g v e -> g v e
+    intersection :: (Hashable v, Eq v, Eq e) => g v e -> g v e -> g v e
 
 
     -- * Transformations
diff --git a/src/Data/Graph/UGraph.hs b/src/Data/Graph/UGraph.hs
--- a/src/Data/Graph/UGraph.hs
+++ b/src/Data/Graph/UGraph.hs
@@ -7,7 +7,7 @@
 module Data.Graph.UGraph where
 
 import qualified Data.Foldable  as F (toList)
-import           Data.List      (foldl')
+import           Data.List      (foldl', intersect)
 import           Data.Semigroup
 import           GHC.Generics   (Generic)
 
@@ -37,11 +37,11 @@
         xs <- readPrec
         return (fromList xs)
 
-instance (Hashable v) => Monoid (UGraph v e) where
+instance (Hashable v, Eq v) => Monoid (UGraph v e) where
     mempty = empty
     mappend = union
 
-instance (Hashable v) => Semigroup (UGraph v e) where
+instance (Hashable v, Eq v) => Semigroup (UGraph v e) where
     (<>) = mappend
 
 instance (Hashable v, Eq v) => Functor (UGraph v) where
@@ -105,10 +105,12 @@
         where go bool v = bool && not (HM.member v $ getLinks v $ unUGraph g)
 
 
-    union = undefined
-    intersection = undefined
-    join = undefined
+    union g1 g2 = insertEdges (toEdgesList g1) $ insertVertices (vertices g1) g2
 
+    intersection g1 g2 =
+        insertVertices (isolatedVertices g1 `intersect` isolatedVertices g2) $
+        fromEdgesList (toEdgesList g1 `intersect` toEdgesList g2)
+
     toList (UGraph _ g) = zip vs $ fmap (\v -> HM.toList $ getLinks v g) vs
         where vs = HM.keys g
 
@@ -194,7 +196,7 @@
 prettyPrint :: (Hashable v, Eq v, Show v, Show e) => UGraph v e -> String
 prettyPrint g =
     "Isolated Vertices: "
-    <> show (filter (\v -> vertexDegree g v == 0) $ vertices g)
+    <> show (isolatedVertices g)
     <> "   "
     <> "Edges: "
     <> show (edges g)