diff --git a/graph/graph.go b/graph/graph.go
index b225c92..7f04060 100644
--- a/graph/graph.go
+++ b/graph/graph.go
@@ -43,6 +43,46 @@ func (e Edge) id() string {
 	return fmt.Sprintf("%q-%q->%q", e.Tail, e.Val, e.Head)
 }
 
+// an edgeIndex maps valueIDs to a set of edgeIDs. Graph keeps two edgeIndex's,
+// one for input edges and one for output edges.
+type edgeIndex map[string]map[string]struct{}
+
+func (ei edgeIndex) cp() edgeIndex {
+	if ei == nil {
+		return edgeIndex{}
+	}
+	ei2 := make(edgeIndex, len(ei))
+	for valID, edgesM := range ei {
+		edgesM2 := make(map[string]struct{}, len(edgesM))
+		for id := range edgesM {
+			edgesM2[id] = struct{}{}
+		}
+		ei2[valID] = edgesM2
+	}
+	return ei2
+}
+
+func (ei edgeIndex) add(valID, edgeID string) {
+	edgesM, ok := ei[valID]
+	if !ok {
+		edgesM = map[string]struct{}{}
+		ei[valID] = edgesM
+	}
+	edgesM[edgeID] = struct{}{}
+}
+
+func (ei edgeIndex) del(valID, edgeID string) {
+	edgesM, ok := ei[valID]
+	if !ok {
+		return
+	}
+
+	delete(edgesM, edgeID)
+	if len(edgesM) == 0 {
+		delete(ei, valID)
+	}
+}
+
 // Graph implements an immutable, unidirectional graph which can hold generic
 // values. All methods are thread-safe as they don't modify the Graph in any
 // way.
@@ -53,11 +93,17 @@ func (e Edge) id() string {
 // Edges are in random order.
 type Graph struct {
 	m map[string]Edge
+
+	// these are indices mapping Value IDs to all the in/out edges for that
+	// Value in the Graph.
+	vIns, vOuts edgeIndex
 }
 
 func (g Graph) cp() Graph {
 	g2 := Graph{
-		m: make(map[string]Edge, len(g.m)),
+		m:     make(map[string]Edge, len(g.m)),
+		vIns:  g.vIns.cp(),
+		vOuts: g.vOuts.cp(),
 	}
 	for id, e := range g.m {
 		g2.m[id] = e
@@ -75,6 +121,8 @@ func (g Graph) AddEdge(e Edge) Graph {
 
 	g2 := g.cp()
 	g2.m[id] = e
+	g2.vIns.add(e.Head.ID, id)
+	g2.vOuts.add(e.Tail.ID, id)
 	return g2
 }
 
@@ -88,6 +136,8 @@ func (g Graph) DelEdge(e Edge) Graph {
 
 	g2 := g.cp()
 	delete(g2.m, id)
+	g2.vIns.del(e.Head.ID, id)
+	g2.vOuts.del(e.Tail.ID, id)
 	return g2
 }
 
@@ -121,14 +171,14 @@ func (g Graph) Edges() []Edge {
 // ValueEdges returns all input (e.Head==v) and output (e.Tail==v) Edges
 // for the given Value in the Graph.
 func (g Graph) ValueEdges(v Value) ([]Edge, []Edge) {
-	var in, out []Edge
-	for _, e := range g.m {
-		if e.Tail.ID == v.ID {
-			out = append(out, e)
-		}
-		if e.Head.ID == v.ID {
-			in = append(in, e)
-		}
+	in := make([]Edge, 0, len(g.vIns[v.ID]))
+	for edgeID := range g.vIns[v.ID] {
+		in = append(in, g.m[edgeID])
+	}
+
+	out := make([]Edge, 0, len(g.vOuts[v.ID]))
+	for edgeID := range g.vOuts[v.ID] {
+		out = append(out, g.m[edgeID])
 	}
 	return in, out
 }