diff --git a/graph/encoding/dot/decode.go b/graph/encoding/dot/decode.go
index f1f5e12f..ccc2868d 100644
--- a/graph/encoding/dot/decode.go
+++ b/graph/encoding/dot/decode.go
@@ -167,21 +167,23 @@ func (gen *generator) addStmt(dst encoding.Builder, stmt ast.Stmt) {
 }
 
 // addEdgeStmt adds the given edge statement to the graph.
-func (gen *generator) addEdgeStmt(dst encoding.Builder, e *ast.EdgeStmt) {
-	fs := gen.addVertex(dst, e.From)
-	ts := gen.addEdge(dst, e.To)
+func (gen *generator) addEdgeStmt(dst encoding.Builder, stmt *ast.EdgeStmt) {
+	fs := gen.addVertex(dst, stmt.From)
+	ts := gen.addEdge(dst, stmt.To)
 	for _, f := range fs {
 		for _, t := range ts {
-			edge, ok := dst.NewEdge(f, t).(encoding.AttributeSetter)
+			edge := dst.NewEdge(f, t)
+			dst.SetEdge(edge)
+			e, ok := edge.(encoding.AttributeSetter)
 			if !ok {
 				continue
 			}
-			for _, attr := range e.Attrs {
+			for _, attr := range stmt.Attrs {
 				a := encoding.Attribute{
 					Key:   attr.Key,
 					Value: attr.Val,
 				}
-				if err := edge.SetAttribute(a); err != nil {
+				if err := e.SetAttribute(a); err != nil {
 					panic(fmt.Errorf("unable to unmarshal edge DOT attribute (%s=%s)", a.Key, a.Value))
 				}
 			}
@@ -216,7 +218,8 @@ func (gen *generator) addEdge(dst encoding.Builder, to *ast.Edge) []graph.Node {
 		ts := gen.addEdge(dst, to.To)
 		for _, f := range fs {
 			for _, t := range ts {
-				dst.NewEdge(f, t)
+				edge := dst.NewEdge(f, t)
+				dst.SetEdge(edge)
 			}
 		}
 	}
diff --git a/graph/encoding/dot/decode_test.go b/graph/encoding/dot/decode_test.go
index 26028116..0416ecd4 100644
--- a/graph/encoding/dot/decode_test.go
+++ b/graph/encoding/dot/decode_test.go
@@ -118,15 +118,9 @@ func (g *dotDirectedGraph) NewNode() graph.Node {
 	return &dotNode{Node: g.DirectedGraph.NewNode()}
 }
 
-// NewEdge adds a new edge from the source to the destination node to the graph,
-// or returns the existing edge if already present.
+// NewEdge returns a new Edge from the source to the destination node.
 func (g *dotDirectedGraph) NewEdge(from, to graph.Node) graph.Edge {
-	if e := g.Edge(from, to); e != nil {
-		return e
-	}
-	e := &dotEdge{Edge: simple.Edge{F: from, T: to}}
-	g.SetEdge(e)
-	return e
+	return &dotEdge{Edge: g.DirectedGraph.NewEdge(from, to)}
 }
 
 // DOTAttributers implements the dot.Attributers interface.
@@ -159,15 +153,9 @@ func (g *dotUndirectedGraph) NewNode() graph.Node {
 	return &dotNode{Node: g.UndirectedGraph.NewNode()}
 }
 
-// NewEdge adds a new edge from the source to the destination node to the graph,
-// or returns the existing edge if already present.
+// NewEdge returns a new Edge from the source to the destination node.
 func (g *dotUndirectedGraph) NewEdge(from, to graph.Node) graph.Edge {
-	if e := g.Edge(from, to); e != nil {
-		return e
-	}
-	e := &dotEdge{Edge: simple.Edge{F: from, T: to}}
-	g.SetEdge(e)
-	return e
+	return &dotEdge{Edge: g.UndirectedGraph.NewEdge(from, to)}
 }
 
 // DOTAttributers implements the dot.Attributers interface.
@@ -222,7 +210,7 @@ func (n *dotNode) Attributes() []encoding.Attribute {
 // dotEdge extends simple.Edge with a label field to test round-trip encoding and
 // decoding of edge DOT label attributes.
 type dotEdge struct {
-	simple.Edge
+	graph.Edge
 	// Edge label.
 	Label string
 }
diff --git a/graph/encoding/encoding.go b/graph/encoding/encoding.go
index b352637c..0f99ac73 100644
--- a/graph/encoding/encoding.go
+++ b/graph/encoding/encoding.go
@@ -10,9 +10,6 @@ import "gonum.org/v1/gonum/graph"
 type Builder interface {
 	graph.Graph
 	graph.Builder
-	// NewEdge adds a new edge from the source to the destination node to the
-	// graph, or returns the existing edge if already present.
-	NewEdge(from, to graph.Node) graph.Edge
 }
 
 // AttributeSetter is implemented by types that can set an encoded graph
diff --git a/graph/encoding/graphql/decode_test.go b/graph/encoding/graphql/decode_test.go
index e713a665..84ba00d8 100644
--- a/graph/encoding/graphql/decode_test.go
+++ b/graph/encoding/graphql/decode_test.go
@@ -163,12 +163,7 @@ func (g *directedGraph) NewNode() graph.Node {
 }
 
 func (g *directedGraph) NewEdge(from, to graph.Node) graph.Edge {
-	if e := g.Edge(from, to); e != nil {
-		return e
-	}
-	e := &edge{Edge: simple.Edge{F: from, T: to}}
-	g.SetEdge(e)
-	return e
+	return &edge{Edge: g.DirectedGraph.NewEdge(from, to)}
 }
 
 type node struct {
@@ -189,7 +184,7 @@ func (n *node) SetIDFromString(uid string) error {
 }
 
 type edge struct {
-	simple.Edge
+	graph.Edge
 	label string
 }
 
diff --git a/graph/graph.go b/graph/graph.go
index 8c284cad..1898d01b 100644
--- a/graph/graph.go
+++ b/graph/graph.go
@@ -93,13 +93,16 @@ type NodeRemover interface {
 	RemoveNode(Node)
 }
 
-// EdgeSetter is an interface for adding edges to a graph.
-type EdgeSetter interface {
+// EdgeAdder is an interface for adding edges to a graph.
+type EdgeAdder interface {
+	// NewEdge returns a new Edge from the source to the destination node.
+	NewEdge(from, to Node) Edge
+
 	// SetEdge adds an edge from one node to another.
 	// If the graph supports node addition the nodes
 	// will be added if they do not exist, otherwise
 	// SetEdge will panic.
-	// The behavior of an EdgeSetter when the IDs
+	// The behavior of an EdgeAdder when the IDs
 	// returned by e.From and e.To are equal is
 	// implementation-dependent.
 	SetEdge(e Edge)
@@ -116,7 +119,7 @@ type EdgeRemover interface {
 // Builder is a graph that can have nodes and edges added.
 type Builder interface {
 	NodeAdder
-	EdgeSetter
+	EdgeAdder
 }
 
 // UndirectedBuilder is an undirected graph builder.
diff --git a/graph/path/internal/testgraphs/shortest.go b/graph/path/internal/testgraphs/shortest.go
index cddb67f1..8d79f939 100644
--- a/graph/path/internal/testgraphs/shortest.go
+++ b/graph/path/internal/testgraphs/shortest.go
@@ -25,7 +25,7 @@ func init() {
 // dynamic shortest path routine in path/dynamic: DStarLite.
 var ShortestPathTests = []struct {
 	Name              string
-	Graph             func() graph.EdgeSetter
+	Graph             func() graph.EdgeAdder
 	Edges             []simple.Edge
 	HasNegativeWeight bool
 	HasNegativeCycle  bool
@@ -40,7 +40,7 @@ var ShortestPathTests = []struct {
 	// Positive weighted graphs.
 	{
 		Name:  "empty directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 
 		Query:  simple.Edge{F: simple.Node(0), T: simple.Node(1)},
 		Weight: math.Inf(1),
@@ -49,7 +49,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "empty undirected",
-		Graph: func() graph.EdgeSetter { return simple.NewUndirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewUndirectedGraph(0, math.Inf(1)) },
 
 		Query:  simple.Edge{F: simple.Node(0), T: simple.Node(1)},
 		Weight: math.Inf(1),
@@ -58,7 +58,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "one edge directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
 		},
@@ -74,7 +74,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "one edge self directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
 		},
@@ -90,7 +90,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "one edge undirected",
-		Graph: func() graph.EdgeSetter { return simple.NewUndirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewUndirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
 		},
@@ -106,7 +106,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "two paths directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node(0), T: simple.Node(2), W: 2},
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -125,7 +125,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "two paths undirected",
-		Graph: func() graph.EdgeSetter { return simple.NewUndirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewUndirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node(0), T: simple.Node(2), W: 2},
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -144,7 +144,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "confounding paths directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->5 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -178,7 +178,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "confounding paths undirected",
-		Graph: func() graph.EdgeSetter { return simple.NewUndirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewUndirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->5 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -212,7 +212,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "confounding paths directed 2-step",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->5 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -247,7 +247,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "confounding paths undirected 2-step",
-		Graph: func() graph.EdgeSetter { return simple.NewUndirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewUndirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->5 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -282,7 +282,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight cycle directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->4 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -306,7 +306,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight cycle^2 directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->4 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -333,7 +333,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight cycle^2 confounding directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->4 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -363,7 +363,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight cycle^3 directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->4 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -393,7 +393,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight 3·cycle^2 confounding directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->4 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -429,7 +429,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight reversed 3·cycle^2 confounding directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			// Add a path from 0->4 of weight 4
 			{F: simple.Node(0), T: simple.Node(1), W: 1},
@@ -465,7 +465,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight |V|·cycle^(n/|V|) directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: func() []simple.Edge {
 			e := []simple.Edge{
 				// Add a path from 0->4 of weight 4
@@ -498,7 +498,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight n·cycle directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: func() []simple.Edge {
 			e := []simple.Edge{
 				// Add a path from 0->4 of weight 4
@@ -531,7 +531,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "zero-weight bi-directional tree with single exit directed",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: func() []simple.Edge {
 			e := []simple.Edge{
 				// Add a path from 0->4 of weight 4
@@ -579,7 +579,7 @@ var ShortestPathTests = []struct {
 	// Negative weighted graphs.
 	{
 		Name:  "one edge directed negative",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node(0), T: simple.Node(1), W: -1},
 		},
@@ -596,7 +596,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "one edge undirected negative",
-		Graph: func() graph.EdgeSetter { return simple.NewUndirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewUndirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node(0), T: simple.Node(1), W: -1},
 		},
@@ -607,7 +607,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "wp graph negative", // http://en.wikipedia.org/w/index.php?title=Johnson%27s_algorithm&oldid=564595231
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node('w'), T: simple.Node('z'), W: 2},
 			{F: simple.Node('x'), T: simple.Node('w'), W: 6},
@@ -630,7 +630,7 @@ var ShortestPathTests = []struct {
 	},
 	{
 		Name:  "roughgarden negative",
-		Graph: func() graph.EdgeSetter { return simple.NewDirectedGraph(0, math.Inf(1)) },
+		Graph: func() graph.EdgeAdder { return simple.NewDirectedGraph(0, math.Inf(1)) },
 		Edges: []simple.Edge{
 			{F: simple.Node('a'), T: simple.Node('b'), W: -2},
 			{F: simple.Node('b'), T: simple.Node('c'), W: -1},
diff --git a/graph/simple/directed.go b/graph/simple/directed.go
index 4172a8ff..1b84f677 100644
--- a/graph/simple/directed.go
+++ b/graph/simple/directed.go
@@ -80,6 +80,11 @@ func (g *DirectedGraph) RemoveNode(n graph.Node) {
 	g.nodeIDs.release(n.ID())
 }
 
+// NewEdge returns a new Edge from the source to the destination node.
+func (g *DirectedGraph) NewEdge(from, to graph.Node) graph.Edge {
+	return &Edge{F: from, T: to}
+}
+
 // SetEdge adds e, an edge from one node to another. If the nodes do not exist, they are added.
 // It will panic if the IDs of the e.From and e.To are equal.
 func (g *DirectedGraph) SetEdge(e graph.Edge) {
diff --git a/graph/simple/undirected.go b/graph/simple/undirected.go
index 37a76e16..3b99ca25 100644
--- a/graph/simple/undirected.go
+++ b/graph/simple/undirected.go
@@ -72,6 +72,11 @@ func (g *UndirectedGraph) RemoveNode(n graph.Node) {
 	g.nodeIDs.release(n.ID())
 }
 
+// NewEdge returns a new Edge from the source to the destination node.
+func (g *UndirectedGraph) NewEdge(from, to graph.Node) graph.Edge {
+	return &Edge{F: from, T: to}
+}
+
 // SetEdge adds e, an edge from one node to another. If the nodes do not exist, they are added.
 // It will panic if the IDs of the e.From and e.To are equal.
 func (g *UndirectedGraph) SetEdge(e graph.Edge) {