diff --git a/src/Algebra/Graph.hs b/src/Algebra/Graph.hs
index ff4327d85..f7d297b86 100644
--- a/src/Algebra/Graph.hs
+++ b/src/Algebra/Graph.hs
@@ -57,6 +57,7 @@ import Control.Applicative (Alternative)
 import Control.DeepSeq (NFData (..))
 import Control.Monad.Compat
 import Data.Foldable (toList)
+import Data.Maybe (fromMaybe)
 import Data.Tree
 
 import Algebra.Graph.Internal
@@ -317,7 +318,7 @@ edges = overlays . map (uncurry edge)
 -- 'isEmpty' . overlays == 'all' 'isEmpty'
 -- @
 overlays :: [Graph a] -> Graph a
-overlays = foldr overlay empty
+overlays = concatg overlay
 
 -- | Connect a given list of graphs.
 -- Complexity: /O(L)/ time and memory, and /O(S)/ size, where /L/ is the length
@@ -331,7 +332,11 @@ overlays = foldr overlay empty
 -- 'isEmpty' . connects == 'all' 'isEmpty'
 -- @
 connects :: [Graph a] -> Graph a
-connects = foldr connect empty
+connects = concatg connect
+
+-- | Auxiliary function, similar to 'mconcat'.
+concatg :: (Graph a -> Graph a -> Graph a) -> [Graph a] -> Graph a
+concatg combine = fromMaybe empty . foldr1Safe combine
 
 -- | Generalised 'Graph' folding: recursively collapse a 'Graph' by applying
 -- the provided functions to the leaves and internal nodes of the expression.
diff --git a/src/Algebra/Graph/Internal.hs b/src/Algebra/Graph/Internal.hs
index eacb3952d..2d83afa22 100644
--- a/src/Algebra/Graph/Internal.hs
+++ b/src/Algebra/Graph/Internal.hs
@@ -20,7 +20,9 @@ module Algebra.Graph.Internal (
     List (..),
 
     -- * Data structures for graph traversal
-    Focus (..), emptyFocus, vertexFocus, overlayFoci, connectFoci, Hit (..)
+    Focus (..), emptyFocus, vertexFocus, overlayFoci, connectFoci, Hit (..),
+
+    foldr1Safe
   ) where
 
 import Prelude ()
@@ -107,3 +109,12 @@ connectFoci x y = Focus (ok x || ok y) (xs <> is y) (os x <> ys) (vs x <> vs y)
 -- | An auxiliary data type for 'hasEdge': when searching for an edge, we can hit
 -- its 'Tail', i.e. the source vertex, the whole 'Edge', or 'Miss' it entirely.
 data Hit = Miss | Tail | Edge deriving (Eq, Ord)
+
+-- | A safe version of 'foldr1'
+foldr1Safe :: (a -> a -> a) -> [a] -> Maybe a
+foldr1Safe f = foldr mf Nothing
+  where
+    mf x m = Just (case m of
+                        Nothing -> x
+                        Just y  -> f x y)
+{-# INLINE foldr1Safe #-}
diff --git a/src/Algebra/Graph/NonEmpty.hs b/src/Algebra/Graph/NonEmpty.hs
index e94946e2e..d2911c452 100644
--- a/src/Algebra/Graph/NonEmpty.hs
+++ b/src/Algebra/Graph/NonEmpty.hs
@@ -279,7 +279,7 @@ connect = Connect
 -- 'vertexSet'   . vertices1 == Set.'Set.fromList' . 'Data.List.NonEmpty.toList'
 -- @
 vertices1 :: NonEmpty a -> NonEmptyGraph a
-vertices1 (x :| xs) = foldr (Overlay . vertex) (vertex x) xs
+vertices1 = overlays1 . fmap vertex
 
 -- | Construct the graph from a list of edges.
 -- Complexity: /O(L)/ time, memory and size, where /L/ is the length of the
@@ -290,7 +290,7 @@ vertices1 (x :| xs) = foldr (Overlay . vertex) (vertex x) xs
 -- 'edgeCount' . edges1   == 'Data.List.NonEmpty.length' . 'Data.List.NonEmpty.nub'
 -- @
 edges1 :: NonEmpty (a, a) -> NonEmptyGraph a
-edges1 (x :| xs) = foldr (Overlay . uncurry edge) (uncurry edge x) xs
+edges1  = overlays1 . fmap (uncurry edge)
 
 -- | Overlay a given list of graphs.
 -- Complexity: /O(L)/ time and memory, and /O(S)/ size, where /L/ is the length
@@ -301,7 +301,7 @@ edges1 (x :| xs) = foldr (Overlay . uncurry edge) (uncurry edge x) xs
 -- overlays1 (x ':|' [y]) == 'overlay' x y
 -- @
 overlays1 :: NonEmpty (NonEmptyGraph a) -> NonEmptyGraph a
-overlays1 = foldr1 overlay
+overlays1 = concatg1 overlay
 
 -- | Connect a given list of graphs.
 -- Complexity: /O(L)/ time and memory, and /O(S)/ size, where /L/ is the length
@@ -312,7 +312,11 @@ overlays1 = foldr1 overlay
 -- connects1 (x ':|' [y]) == 'connect' x y
 -- @
 connects1 :: NonEmpty (NonEmptyGraph a) -> NonEmptyGraph a
-connects1 = foldr1 connect
+connects1 = concatg1 connect
+
+-- | Auxiliary function, similar to 'sconcat'.
+concatg1 :: (NonEmptyGraph a -> NonEmptyGraph a -> NonEmptyGraph a) -> NonEmpty (NonEmptyGraph a) -> NonEmptyGraph a
+concatg1 combine (x :| xs) = maybe x (combine x) $ foldr1Safe combine xs
 
 -- | Generalised graph folding: recursively collapse a 'NonEmptyGraph' by
 -- applying the provided functions to the leaves and internal nodes of the