diff --git a/IntGraph.cabal b/IntGraph.cabal
--- a/IntGraph.cabal
+++ b/IntGraph.cabal
@@ -2,10 +2,10 @@
 --
 -- see: https://github.com/sol/hpack
 --
--- hash: bf685344ee77315d99fcd243d23b7e410a495e791dfd796f1ce4c76491f45501
+-- hash: c91eea0f55615780e544e99395b8d3f7208a7f28ccd97490f760ea25f3b12017
 
 name:           IntGraph
-version:        0.1.0.0
+version:        0.1.1.0
 synopsis:       Dynamically sized graph library
 description:    Graph implemented as an IntMap to Sets of Ints. Functions for Directed and Undirected graphs are provided.
 category:       Data structure
@@ -35,7 +35,9 @@
     , containers
   exposed-modules:
       Data.IntGraph.Directed
+      Data.IntGraph.Directed.Algorithms
       Data.IntGraph.Undirected
+      Data.IntGraph.Undirected.Algorithms
   other-modules:
       Data.IntGraph
       Paths_IntGraph
diff --git a/Setup.hs b/Setup.hs
new file mode 100644
--- /dev/null
+++ b/Setup.hs
@@ -0,0 +1,2 @@
+import Distribution.Simple
+main = defaultMain
diff --git a/src/Data/IntGraph.hs b/src/Data/IntGraph.hs
--- a/src/Data/IntGraph.hs
+++ b/src/Data/IntGraph.hs
@@ -9,19 +9,26 @@
   empty,
   addNode,
   nodes,
-  removeNode)
+  removeNode,
+  null,
+  nullEdges)
   where
 
+import Prelude hiding (null)
+
 import qualified Data.IntMap.Strict as I
-import           Data.IntMap.Strict    (IntMap)
-import qualified Data.Set           as S
-import           Data.Set              (Set)
+import           Data.IntMap.Strict (IntMap)
+import qualified Data.IntSet        as S
+import           Data.IntSet        (IntSet)
 
+import Data.Semigroup
+import Data.Monoid
+
 -- | The nodes of the graph are Ints
 type Node = Int
 
--- | A NodeSet is a Set of Nodes
-type NodeSet = Set Node
+-- | A NodeSet is a IntSet of Nodes
+type NodeSet = IntSet
 
 -- | An edge is a pair of nodes
 type Edge = (Node, Node)
@@ -45,3 +52,26 @@
 
 -- | the empty graph
 empty = IG I.empty
+
+-- | Is the graph empty
+null :: IntGraph -> Bool
+null (IG graph) = I.null graph
+
+-- | Is the edge set empty
+nullEdges :: IntGraph -> Bool
+nullEdges (IG graph) = I.null $ I.filter (not . S.null) graph
+
+-- | Joins to graphs by unioning the vertex and edge sets
+overlay :: IntGraph -> IntGraph -> IntGraph
+overlay (IG g1) (IG g2) = IG $ I.unionWith (S.union) g1 g2
+
+instance Eq IntGraph where
+  (IG g1) == (IG g2) = g1 == g2
+
+instance Semigroup IntGraph where
+  (<>) = overlay
+
+instance Monoid IntGraph where
+  mempty = empty
+
+  mappend = overlay
diff --git a/src/Data/IntGraph/Directed.hs b/src/Data/IntGraph/Directed.hs
--- a/src/Data/IntGraph/Directed.hs
+++ b/src/Data/IntGraph/Directed.hs
@@ -12,11 +12,15 @@
   addEdge,
   edges,
   removeEdge,
-  fromEdges) 
+  fromEdges,
+  null,
+  nullEdges) 
   where
 
+import Prelude hiding (null)
+
 import qualified Data.IntMap as I
-import qualified Data.Set    as S
+import qualified Data.IntSet as S
 
 import Data.IntGraph
 
diff --git a/src/Data/IntGraph/Directed/Algorithms.hs b/src/Data/IntGraph/Directed/Algorithms.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/IntGraph/Directed/Algorithms.hs
@@ -0,0 +1,32 @@
+module Data.IntGraph.Directed.Algorithms where
+
+import Prelude hiding (null)
+
+import Control.Applicative ((<|>))
+import Data.Maybe (isJust, fromJust)
+
+import Data.IntGraph.Directed
+
+
+-- | If there is a vertex cover of size k, returns the vertex cover
+-- | Otherwise Nothing
+-- | O(n*2^k)
+vertexCoverDec :: IntGraph -> Int -> Maybe [Node]
+vertexCoverDec graph k
+  | nullEdges graph = Just []
+  | k == 0 = Nothing
+  | otherwise = uCover <|> vCover
+  where
+    (u, v) = head $ edges graph
+    uCover = (u :) <$> vertexCoverDec (removeNode u graph) (k - 1)
+    vCover = (v :) <$> vertexCoverDec (removeNode v graph) (k - 1)
+
+-- | vertexCoverDec, but just returns true or false
+vertexCoverBool :: IntGraph -> Int -> Bool
+vertexCoverBool = (isJust .) . vertexCoverDec
+
+-- | Optimal vertex cover
+-- | Not all that efficient, just keeps trying difference values for k
+vertexCover :: IntGraph -> [Node]
+vertexCover graph = let f k = vertexCoverDec graph k <|> f (k + 1)
+                     in fromJust $ f 0
diff --git a/src/Data/IntGraph/Undirected.hs b/src/Data/IntGraph/Undirected.hs
--- a/src/Data/IntGraph/Undirected.hs
+++ b/src/Data/IntGraph/Undirected.hs
@@ -12,11 +12,15 @@
   addEdge,
   edges,
   removeEdge,
-  fromEdges)
+  fromEdges,
+  null,
+  nullEdges)
   where
 
+import Prelude hiding (null)
+
 import qualified Data.IntMap as I
-import qualified Data.Set    as S
+import qualified Data.IntSet as S
 import           Data.List   (delete)
 
 import           Data.IntGraph
diff --git a/src/Data/IntGraph/Undirected/Algorithms.hs b/src/Data/IntGraph/Undirected/Algorithms.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/IntGraph/Undirected/Algorithms.hs
@@ -0,0 +1,32 @@
+module Data.IntGraph.Undirected.Algorithms where
+
+import Prelude hiding (null)
+
+import Control.Applicative ((<|>))
+import Data.Maybe (isJust, fromJust)
+
+import Data.IntGraph.Undirected
+
+
+-- | If there is a vertex cover of size k, returns the vertex cover
+-- | Otherwise Nothing
+-- | O(n*2^k)
+vertexCoverDec :: IntGraph -> Int -> Maybe [Node]
+vertexCoverDec graph k
+  | nullEdges graph = Just []
+  | k == 0 = Nothing
+  | otherwise = uCover <|> vCover
+  where
+    (u, v) = head $ edges graph
+    uCover = (u :) <$> vertexCoverDec (removeNode u graph) (k - 1)
+    vCover = (v :) <$> vertexCoverDec (removeNode v graph) (k - 1)
+
+-- | vertexCoverDec, but just returns true or false
+vertexCoverBool :: IntGraph -> Int -> Bool
+vertexCoverBool = (isJust .) . vertexCoverDec
+
+-- | Optimal vertex cover
+-- | Not all that efficient, just keeps trying difference values for k
+vertexCover :: IntGraph -> [Node]
+vertexCover graph = let f k = vertexCoverDec graph k <|> f (k + 1)
+                     in fromJust $ f 0