From 00174ae81df2a2838bc12280fae069526321be74 Mon Sep 17 00:00:00 2001
From: Pat Garrity <pfm@garrity.co>
Date: Fri, 5 Jun 2026 21:30:28 -0500
Subject: [PATCH] WIP on a refactor and massive set of improvements

---
 .../main/scala/gs/graph/v0/Adjacency.scala    | 376 +++++++++++++++---
 .../src/main/scala/gs/graph/v0/Edge.scala     |  13 +
 .../src/main/scala/gs/graph/v0/Graph.scala    |   5 -
 .../scala/gs/graph/v0/GraphException.scala    |  18 +
 .../scala/gs/graph/v0/GraphTraversal.scala    |  95 +++--
 .../scala/gs/graph/v0/UndirectedGraph.scala   |  19 -
 .../main/scala/gs/graph/v0/data/DataDag.scala |  12 +-
 .../scala/gs/graph/v0/data/DataDigraph.scala  |  12 +-
 .../main/scala/gs/graph/v0/directed/Dag.scala |  12 +-
 .../scala/gs/graph/v0/directed/Digraph.scala  |  46 +--
 .../graph/v0/directed/SingleRootDigraph.scala | 144 -------
 .../directed/StronglyConnectedComponent.scala |  77 ++--
 .../gs/graph/v0/UndirectedGraphTests.scala    |   3 -
 13 files changed, 476 insertions(+), 356 deletions(-)
 delete mode 100644 modules/core/src/main/scala/gs/graph/v0/directed/SingleRootDigraph.scala

diff --git a/modules/core/src/main/scala/gs/graph/v0/Adjacency.scala b/modules/core/src/main/scala/gs/graph/v0/Adjacency.scala
index fcce9de..471322c 100644
--- a/modules/core/src/main/scala/gs/graph/v0/Adjacency.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/Adjacency.scala
@@ -1,14 +1,32 @@
 package gs.graph.v0
 
+import scala.collection.IndexedSeqView
 import scala.collection.mutable.ListBuffer
 
-/** An immutable adjacency list representation.
-  *
-  * The index of the collection is the "from" [[Vertex]], and each element of
-  * the corresponding vector is a "to" [[Vertex]] -- there are edges _from_ some
-  * vertex _to_ another vertex.
+/** An adjacency list representation.
   */
-final class Adjacency private (val neighbors: Vector[Vector[Vertex]]):
+sealed trait Adjacency:
+  /** @return
+    *   The number of vertices in this adjacency list.
+    */
+  def numberOfVertices: Size
+
+  /** @return
+    *   The number of edges in this adjacency list.
+    */
+  def numberOfEdges: Size
+
+  /** @return
+    *   View over the vertices in this adjacency.
+    */
+  def vertices: IndexedSeqView[Vertex]
+
+  /** @return
+    *   The list of entries in this adjacency. Each entry is a vertex paired
+    *   with the neighbors of that vertex.
+    */
+  def entries: Map[Vertex, Vector[Vertex]]
+
   /** Get the vector of [[Vertex]] that receive a connection _from_ the input
     * [[Vertex]].
     *
@@ -18,7 +36,16 @@ final class Adjacency private (val neighbors: Vector[Vector[Vertex]]):
     *   The vector of [[Vertex]] that receive a connection _from_ the input
     *   [[Vertex]].
     */
-  def at(vertex: Vertex): Vector[Vertex] = neighbors(vertex.ordinal)
+  def at(vertex: Vertex): Vector[Vertex]
+
+  /** Determine if the given [[Vertex]] is represented within this adjacency.
+    *
+    * @param vertex
+    *   The [[Vertex]] to investigate.
+    * @return
+    *   `True` if the [[Vertex]] is contained, `false` otherwise.
+    */
+  def contains(vertex: Vertex): Boolean
 
   /** Get the vector of _incoming_ [[Vertex]] that point _to_ some input
     * [[Vertex]].
@@ -28,57 +55,47 @@ final class Adjacency private (val neighbors: Vector[Vector[Vertex]]):
     * @return
     *   The list of [[Vertex]] that have an edge _to_ the input [[Vertex]].
     */
-  def incoming(vertex: Vertex): Vector[Vertex] =
-    if vertex.ordinal >= neighbors.length then Vector.empty
-    else
-      neighbors.zipWithIndex
-        .filter {
-          // ignore the neighbors of the input vertex
-          case (_, index) => index != vertex.ordinal
-        }
-        .filter(_._1.contains(vertex))
-        .map(_._2)
-        .map(Vertex(_))
+  def incoming(vertex: Vertex): Vector[Vertex]
 
   /** Express this [[Adjacency]] as a vector of [[Edge]].
     *
     * @return
     *   The vector of [[Edge]] derived from this [[Adjacency]].
     */
-  def toEdges(): Vector[Edge] =
-    neighbors.zipWithIndex.flatMap { case (tos, index) =>
-      val from = Vertex(index)
-      tos.map(to => Edge(from, to)).distinct
-    }
+  def toEdges(): Vector[Edge]
 
-  /** The number of vertices represented by this adjacency list.
-    */
-  lazy val numberOfVertices: Size = Size.fromVector(neighbors)
-
-  /** Perform a linear traversal for each [[gs.graph.v0.Vertex]] to calculate
-    * the total number of edges in this adjacency list.
+  /** Extract a subset of vertices and their neighbors, such that the neighbors
+    * are defined within the subset.
     *
+    * Throws an exception if any input [[Vertex]] is not a member of this
+    * adjacency.
+    *
+    * @param vertices
+    *   The subset of [[Vertex]].
     * @return
-    *   The number of edges in this adjacency list.
+    *   The subset adjacency.
     */
-  lazy val numberOfEdges: Size = Size(neighbors.map(_.length).reduce(_ + _))
+  def subset(vertices: Set[Vertex]): Adjacency =
+    val notContained = vertices.filterNot(contains)
+    if notContained.nonEmpty then
+      throw GraphException.AdjacencyDoesNotContainVertices(
+        vertices,
+        numberOfVertices
+      )
+    else
+      MappedAdjacency.validate(vertices.map { vertex =>
+        // Only take the neighbors that are represented within the subset.
+        vertex -> at(vertex).filter(vertices.contains)
+      }.toMap)
 
-  /** All vertices that do not have inbound connections.
+  /** @inheritDocs
     */
-  lazy val roots: Vector[Vertex] =
-    val counts = Array.fill(neighbors.length)(0)
-    neighbors.foreach { ns =>
-      // Each vertex listed here is receiving an inbound connection, if we
-      // assume the graph is directed.
-      ns.foreach(v => counts(v.ordinal) += 1)
-    }
-    counts.filter(_ == 0).map(Vertex(_)).toVector
-
   override def equals(that: Any): Boolean =
-    that match
-      case other: Adjacency =>
-        neighbors.length == other.neighbors.length
-        && neighbors
+    val tuple = (this, that)
+    tuple match
+      case (me: DefaultAdjacency, other: DefaultAdjacency) =>
+        me.neighbors.length == other.neighbors.length
+        && me.neighbors
           .zip(other.neighbors)
           .map { case (x, y) => x.sameElements(y) }
           .forall(result => result)
@@ -86,6 +103,8 @@ final class Adjacency private (val neighbors: Vector[Vector[Vertex]]):
 
 object Adjacency:
 
+  given CanEqual[Adjacency, Adjacency] = CanEqual.derived
+
   /** Instantiate a new [[Adjacency]] list from a raw representation.
     *
     * @param adj
@@ -93,7 +112,8 @@ object Adjacency:
     * @return
     *   The new [[Adjacency]].
     */
-  def apply(adj: Vector[Vector[Vertex]]): Adjacency = new Adjacency(adj)
+  def apply(adj: Vector[Vector[Vertex]]): Adjacency =
+    DefaultAdjacency.apply(adj)
 
   /** Create an empty adjacency list for some number of vertices. No vertex has
     * any connections to another vertex.
@@ -104,19 +124,156 @@ object Adjacency:
     *   Some new empty adjacency.
     */
   def noEdges(numberOfVertices: Size): Adjacency =
-    new Adjacency(Vector.fill(numberOfVertices.value)(Vector.empty))
-
-  given CanEqual[Adjacency, Adjacency] = CanEqual.derived
+    DefaultAdjacency.noEdges(numberOfVertices)
 
   /** @return
     *   The empty [[Adjacency]] list.
     */
-  final val Empty: Adjacency = new Adjacency(Vector.empty)
+  final val Empty: Adjacency = DefaultAdjacency.Empty
 
   /** @return
     *   The single-vertex [[Adjacency]] list.
     */
-  final val Single: Adjacency = new Adjacency(Vector(Vector.empty))
+  final val Single: Adjacency = DefaultAdjacency.Single
+
+  /** Calculate an [[Adjacency]] from some collection of [[Edge]].
+    *
+    * @param numberOfVertices
+    *   The number of [[Vertex]] (`N`) in this graph.
+    * @param edges
+    *   The collection of [[Edge]] present in this graph.
+    * @return
+    *   The calculated [[Adjacency]].
+    */
+  def fromEdges(
+    numberOfVertices: Size,
+    edges: Iterable[Edge]
+  ): Adjacency =
+    DefaultAdjacency.fromEdges(numberOfVertices, edges)
+
+  /** Calculate an [[Adjacency]] from some collection of [[Edge]].
+    *
+    * @param edges
+    *   The collection of [[Edge]] present in this graph.
+    * @return
+    *   The calculated [[Adjacency]].
+    */
+  def fromEdges(
+    edges: Iterable[Edge]
+  ): Adjacency =
+    DefaultAdjacency.fromEdges(edges)
+
+end Adjacency
+
+/** An immutable adjacency list representation.
+  *
+  * This implementation uses a 0-based indexing system, where each vertex has an
+  * index from 0 to N-1. See [[MappedAdjacency]] for an implementation that does
+  * not require a complete ordered set.
+  *
+  * The index of the collection is the "from" [[Vertex]], and each element of
+  * the corresponding vector is a "to" [[Vertex]] -- there are edges _from_ some
+  * vertex _to_ another vertex.
+  *
+  * @param numberOfVertices
+  *   The number of vertices represented by this [[Adjacency]].
+  * @param neighbors
+  *   The complete set of relationships.
+  */
+final class DefaultAdjacency private (
+  val numberOfVertices: Size,
+  val neighbors: Vector[Vector[Vertex]]
+) extends Adjacency:
+
+  /** @inheritDocs
+    */
+  override def vertices: IndexedSeqView[Vertex] =
+    (0 until numberOfVertices.value).view.map(Vertex(_))
+
+  /** @inheritDocs
+    */
+  override def entries: Map[Vertex, Vector[Vertex]] =
+    neighbors.zipWithIndex.map { case (ns, index) => Vertex(index) -> ns }.toMap
+
+  /** @inheritDocs
+    */
+  override def at(vertex: Vertex): Vector[Vertex] =
+    if vertex < numberOfVertices then neighbors(vertex.ordinal)
+    else Vector.empty
+
+  /** @inheritDocs
+    */
+  override def contains(vertex: Vertex): Boolean =
+    vertex < numberOfVertices
+
+  /** @inheritDocs
+    */
+  override def incoming(vertex: Vertex): Vector[Vertex] =
+    if vertex >= numberOfVertices then Vector.empty
+    else
+      neighbors.zipWithIndex
+        .filter {
+          // ignore the neighbors of the input vertex
+          // select those that reference the input
+          case (values, index) =>
+            (index != vertex.ordinal) && values.contains(vertex)
+        }
+        .map(_._2)
+        .map(Vertex(_))
+
+  /** @inheritDocs
+    */
+  override def toEdges(): Vector[Edge] =
+    neighbors.zipWithIndex.flatMap { case (tos, index) =>
+      val from = Vertex(index)
+      tos.map(to => Edge(from, to)).distinct
+    }
+
+  /** @inheritDocs
+    */
+  override lazy val numberOfEdges: Size = Size(
+    neighbors.map(_.length).reduce(_ + _)
+  )
+
+object DefaultAdjacency:
+
+  /** Instantiate a new [[Adjacency]] list from a raw representation.
+    *
+    * @param adj
+    *   The underlying vector of values.
+    * @return
+    *   The new [[Adjacency]].
+    */
+  def apply(adj: Vector[Vector[Vertex]]): Adjacency =
+    new DefaultAdjacency(
+      numberOfVertices = Size.fromVector(adj),
+      neighbors = adj
+    )
+
+  /** Create an empty adjacency list for some number of vertices. No vertex has
+    * any connections to another vertex.
+    *
+    * @param numberOfVertices
+    *   The number of vertices in this disconnected graph.
+    * @return
+    *   Some new empty adjacency.
+    */
+  def noEdges(numberOfVertices: Size): Adjacency =
+    new DefaultAdjacency(
+      numberOfVertices = numberOfVertices,
+      neighbors = Vector.fill(numberOfVertices.value)(Vector.empty)
+    )
+
+  /** @return
+    *   The empty [[Adjacency]] list.
+    */
+  final val Empty: Adjacency = new DefaultAdjacency(Size.Zero, Vector.empty)
+
+  /** @return
+    *   The single-vertex [[Adjacency]] list.
+    */
+  final val Single: Adjacency =
+    new DefaultAdjacency(Size.One, Vector(Vector.empty))
 
   /** Calculate an [[Adjacency]] from some collection of [[Edge]].
     *
@@ -140,7 +297,7 @@ object Adjacency:
       else
         val _ = buffs(edge.from.ordinal).addOne(edge.to)
     }
-    new Adjacency(buffs.map(_.distinct.toVector))
+    new DefaultAdjacency(numberOfVertices, buffs.map(_.distinct.toVector))
 
   /** Calculate an [[Adjacency]] from some collection of [[Edge]].
     *
@@ -157,7 +314,7 @@ object Adjacency:
     val _     = edges.foreach { edge =>
       val _ = buffs(edge.from.ordinal).addOne(edge.to)
     }
-    new Adjacency(buffs.map(_.distinct.toVector))
+    new DefaultAdjacency(Size(numberOfVertices), buffs.map(_.distinct.toVector))
 
   private def findMaximumVertex(edges: Iterable[Edge]): Vertex =
     var maximum = Vertex.Zero
@@ -167,4 +324,115 @@ object Adjacency:
     }
     maximum
 
-end Adjacency
+end DefaultAdjacency
+
+/** An immutable adjacency list implementation.
+  *
+  * This implementation maps [[Vertex]] values to lists of their neighbors.
+  *
+  * @param numberOfVertices
+  *   The number of vertices contained in the adjacency.
+  * @param neighbors
+  *   The list of neighbors for each vertex in the adjacency.
+  */
+final class MappedAdjacency private (
+  val numberOfVertices: Size,
+  val neighbors: Map[Vertex, Vector[Vertex]]
+) extends Adjacency:
+
+  /** @inheritDocs
+    */
+  override def vertices: IndexedSeqView[Vertex] =
+    neighbors.keySet.toIndexedSeq.view
+
+  /** @inheritDocs
+    */
+  override def entries: Map[Vertex, Vector[Vertex]] =
+    neighbors
+
+  /** @inheritDocs
+    */
+  override lazy val numberOfEdges: Size =
+    Size(neighbors.map(_._2.length).reduce(_ + _))
+
+  /** @inheritDocs
+    */
+  override def at(vertex: Vertex): Vector[Vertex] =
+    neighbors.get(vertex).getOrElse(Vector.empty)
+
+  /** @inheritDocs
+    */
+  override def contains(vertex: Vertex): Boolean =
+    neighbors.contains(vertex)
+
+  /** @inheritDocs
+    */
+  override def incoming(vertex: Vertex): Vector[Vertex] =
+    neighbors.view
+      .filter {
+        // ignore the neighbors of the input vertex
+        // select those that reference the input
+        case (key, values) => (key != vertex) && values.contains(vertex)
+      }
+      .map(_._1)
+      .toVector
+
+  /** @inheritDocs
+    */
+  override def toEdges(): Vector[Edge] =
+    neighbors.view.flatMap { case (from, tos) =>
+      tos.map(to => Edge(from, to)).distinct
+    }.toVector
+
+object MappedAdjacency:
+
+  final val Empty: Adjacency = new MappedAdjacency(Size.Zero, Map.empty)
+
+  def single(vertex: Vertex): Adjacency =
+    new MappedAdjacency(Size.One, Map(vertex -> Vector.empty))
+
+  /** Validate the given mapped neighbors. Enforces that all neighbors of some
+    * vertex _must_ be keys of the map -- neighbors can only be vertices that
+    * are part of the same graph.
+    *
+    * Throws an exception if the input is invalid.
+    *
+    * @param neighbors
+    *   The mapped neighbors.
+    * @return
+    *   The new [[Adjacency]].
+    */
+  def validate(
+    neighbors: Map[Vertex, Vector[Vertex]]
+  ): Adjacency =
+    val vertices = neighbors.keySet
+    val _        = neighbors.foreach { case (v, ns) =>
+      ns.foreach { candidate =>
+        if !vertices.contains(candidate) then
+          throw GraphException.NeighborNotContainedByAdjacency(
+            candidate = candidate,
+            vertex = v
+          )
+        else ()
+      }
+    }
+    new MappedAdjacency(Size.of(neighbors), neighbors)
+
+  /** Instantiate a new [[MappedAdjacency]] based on the given edges.
+    *
+    * @param edges
+    *   The edges that represent the adjacency.
+    * @return
+    *   the new adjacency instance.
+    */
+  def fromEdges(
+    edges: Iterable[Edge]
+  ): Adjacency =
+    val acc = scala.collection.mutable.Map.empty[Vertex, ListBuffer[Vertex]]
+    edges.foreach { edge =>
+      acc.getOrElseUpdate(edge.from, ListBuffer.empty).addOne(edge.to)
+    }
+    val neighbors = acc.map { case (v, ns) => v -> ns.toVector }.toMap
+    new MappedAdjacency(Size.of(neighbors), neighbors)
+
+end MappedAdjacency
diff --git a/modules/core/src/main/scala/gs/graph/v0/Edge.scala b/modules/core/src/main/scala/gs/graph/v0/Edge.scala
index bd3ab5b..8ec31c0 100644
--- a/modules/core/src/main/scala/gs/graph/v0/Edge.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/Edge.scala
@@ -105,6 +105,19 @@ object Edge:
     v2: Int
   ): Edge = Edge(Vertex(v1), Vertex(v2))
 
+  /** Instantiate a new Edge.
+    *
+    * Throws an exception if the given vertices are the same or are not valid
+    * [[Vertex]] values.
+    *
+    * @param pair
+    *   The pair to convert to an edge.
+    * @return
+    *   The new edge.
+    */
+  def fromPair(pair: (Int, Int)): Edge =
+    Edge(Vertex(pair._1), Vertex(pair._2))
+
   /** Instantiate a list of Edge.
     *
     * @param edges
diff --git a/modules/core/src/main/scala/gs/graph/v0/Graph.scala b/modules/core/src/main/scala/gs/graph/v0/Graph.scala
index acd3e3c..f5a7258 100644
--- a/modules/core/src/main/scala/gs/graph/v0/Graph.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/Graph.scala
@@ -46,11 +46,6 @@ trait Graph:
     */
   def isSingle: Boolean = numberOfVertices == Size.One
 
-  /** @return
-    *   The roots that should be used for traversal operations.
-    */
-  def selectRoots(): Vector[Vertex]
-
   /** Get the neighbors for any given vertex.
     *
     * @param vertex
diff --git a/modules/core/src/main/scala/gs/graph/v0/GraphException.scala b/modules/core/src/main/scala/gs/graph/v0/GraphException.scala
index 0f43543..ff4a8fb 100644
--- a/modules/core/src/main/scala/gs/graph/v0/GraphException.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/GraphException.scala
@@ -31,4 +31,22 @@ object GraphException:
           but that root exceeds the graph bound of '$numberOfVertices'
           vertices.""".stripMargin
 
+  case class AdjacencyDoesNotContainVertices(
+    vertices: Iterable[Vertex],
+    numberOfVertices: Size
+  ) extends GraphException:
+
+    override def getMessage(): String =
+      s"""The following vertices were not found within an Adjacency of size
+          '$numberOfVertices': [${vertices.mkString(", ")}]""".stripMargin
+
+  case class NeighborNotContainedByAdjacency(
+    candidate: Vertex,
+    vertex: Vertex
+  ) extends GraphException:
+
+    override def getMessage(): String =
+      s"""Attempted to reference neighbor '$candidate' of vertex '$vertex', but
+          '$candidate' is not a valid vertex in the adjacency.""".stripMargin
+
 end GraphException
diff --git a/modules/core/src/main/scala/gs/graph/v0/GraphTraversal.scala b/modules/core/src/main/scala/gs/graph/v0/GraphTraversal.scala
index 56f707c..0b1d407 100644
--- a/modules/core/src/main/scala/gs/graph/v0/GraphTraversal.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/GraphTraversal.scala
@@ -159,16 +159,18 @@ object GraphTraversal:
     val s          = Stack.empty[Vertex]
     val discovered = Array.fill(graph.numberOfVertices.value)(false)
 
-    graph.selectRoots().foreach { root =>
-      val _ = s.push(root)
-      while !s.isEmpty
-      do
-        val v = s.pop()
-        if !discovered(v.ordinal) then
-          val _ = output.addOne(visit(v, graph.data(v.ordinal)))
-          discovered(v.ordinal) = true
-          graph.neighbors(v).foreach(w => s.push(w))
-        else ()
+    graph.view.foreach { root =>
+      if !discovered(root.ordinal) then
+        val _ = s.push(root)
+        while !s.isEmpty
+        do
+          val v = s.pop()
+          if !discovered(v.ordinal) then
+            val _ = output.addOne(visit(v, graph.data(v.ordinal)))
+            discovered(v.ordinal) = true
+            val _ = graph.neighbors(v).reverseIterator.foreach(w => s.push(w))
+          else ()
+      else ()
     }
 
     output.toList
@@ -283,20 +285,24 @@ object GraphTraversal:
   ): Unit =
     val q       = Queue.empty[Vertex]
     val visited = Array.fill(graph.numberOfVertices.value)(false)
-    val _       = graph.selectRoots().foreach(q.enqueue)
 
-    while !q.isEmpty
-    do
-      val v = q.dequeue()
-      if !visited(v.ordinal) then
-        val _ = visit(v, graph.data(v.ordinal))
-        visited(v.ordinal) = true
-        graph.neighbors(v).foreach { neighbor =>
-          if !visited(neighbor.ordinal) then q.enqueue(neighbor)
+    graph.view.foreach { root =>
+      if !visited(root.ordinal) then
+        val _ = q.enqueue(root)
+        while !q.isEmpty
+        do
+          val v = q.dequeue()
+          if !visited(v.ordinal) then
+            val _ = visit(v, graph.data(v.ordinal))
+            visited(v.ordinal) = true
+            graph.neighbors(v).foreach { neighbor =>
+              if !visited(neighbor.ordinal) then q.enqueue(neighbor)
+              else ()
+            }
           else ()
-        }
+        ()
       else ()
-    ()
+    }
 
   def bfs[A, Out](
     graph: AnyGraphWithData[A],
@@ -305,19 +311,23 @@ object GraphTraversal:
     val output  = ListBuffer.empty[Out]
     val q       = Queue.empty[Vertex]
     val visited = Array.fill(graph.numberOfVertices.value)(false)
-    val _       = graph.selectRoots().foreach(q.enqueue)
 
-    while !q.isEmpty
-    do
-      val v = q.dequeue()
-      if !visited(v.ordinal) then
-        val _ = output.addOne(visit(v, graph.data(v.ordinal)))
-        visited(v.ordinal) = true
-        graph.neighbors(v).foreach { neighbor =>
-          if !visited(neighbor.ordinal) then q.enqueue(neighbor)
+    graph.view.foreach { root =>
+      if !visited(root.ordinal) then
+        val _ = q.enqueue(root)
+        while !q.isEmpty
+        do
+          val v = q.dequeue()
+          if !visited(v.ordinal) then
+            val _ = output.addOne(visit(v, graph.data(v.ordinal)))
+            visited(v.ordinal) = true
+            graph.neighbors(v).foreach { neighbor =>
+              if !visited(neighbor.ordinal) then q.enqueue(neighbor)
+              else ()
+            }
           else ()
-        }
       else ()
+    }
 
     output.toList
 
@@ -329,19 +339,22 @@ object GraphTraversal:
     var acc     = initial
     val q       = Queue.empty[Vertex]
     val visited = Array.fill(graph.numberOfVertices.value)(false)
-    val _       = graph.selectRoots().foreach(q.enqueue)
 
-    while !q.isEmpty
-    do
-      val v = q.dequeue()
-      if !visited(v.ordinal) then
-        acc = f(acc, graph.data(v.ordinal))
-        visited(v.ordinal) = true
-        graph.neighbors(v).foreach { neighbor =>
-          if !visited(neighbor.ordinal) then q.enqueue(neighbor)
+    graph.view.foreach { root =>
+      if !visited(root.ordinal) then
+        while !q.isEmpty
+        do
+          val v = q.dequeue()
+          if !visited(v.ordinal) then
+            acc = f(acc, graph.data(v.ordinal))
+            visited(v.ordinal) = true
+            graph.neighbors(v).foreach { neighbor =>
+              if !visited(neighbor.ordinal) then q.enqueue(neighbor)
+              else ()
+            }
           else ()
-        }
       else ()
+    }
 
     acc
 
diff --git a/modules/core/src/main/scala/gs/graph/v0/UndirectedGraph.scala b/modules/core/src/main/scala/gs/graph/v0/UndirectedGraph.scala
index 4e52df4..dd3d87c 100644
--- a/modules/core/src/main/scala/gs/graph/v0/UndirectedGraph.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/UndirectedGraph.scala
@@ -15,25 +15,6 @@ class UndirectedGraph(
     */
   final override val disposition: GraphDisposition = GraphDisposition.Undirected
 
-  /** @inheritDocs
-    */
-  override def selectRoots(): Vector[Vertex] = roots
-
-  /** The lazily calculated roots of this graph.
-    *
-    * In an undirected graph, the roots are calculated by selecting an arbitrary
-    * vertex from each connected component in the graph.
-    */
-  lazy val roots: Vector[Vertex] =
-    calculateRoots()
-
-  private def calculateRoots(): Vector[Vertex] =
-    if numberOfVertices == Size.Zero then Vector.empty
-    else oneFromEachConnectedComponent().toVector
-
-  private def oneFromEachConnectedComponent(): List[Vertex] =
-    ConnectedComponent.findAll(this).map(_.root)
-
 end UndirectedGraph
 
 object UndirectedGraph:
diff --git a/modules/core/src/main/scala/gs/graph/v0/data/DataDag.scala b/modules/core/src/main/scala/gs/graph/v0/data/DataDag.scala
index 2899a5c..400fc19 100644
--- a/modules/core/src/main/scala/gs/graph/v0/data/DataDag.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/data/DataDag.scala
@@ -3,7 +3,6 @@ package gs.graph.v0.data
 import gs.graph.v0.Adjacency
 import gs.graph.v0.GraphException
 import gs.graph.v0.Size
-import gs.graph.v0.Vertex
 import gs.graph.v0.directed.Dag
 
 /** Specialization of [[Dag]] that associates _data_ with _each [[Vertex]]_.
@@ -11,9 +10,8 @@ import gs.graph.v0.directed.Dag
 final class DataDag[A] private (
   val data: Vector[A],
   n: Size,
-  a: Adjacency,
-  r: Vector[Vertex]
-) extends Dag(n, a, r)
+  a: Adjacency
+) extends Dag(n, a)
       with AnyGraphWithData[A]
 
 object DataDag:
@@ -25,8 +23,7 @@ object DataDag:
     new DataDag[A](
       Vector.empty,
       Size.Zero,
-      Adjacency.Empty,
-      Vector.empty
+      Adjacency.Empty
     )
 
   /** Instantiate a new [[DataDag]] given an input [[Dag]] and some data.
@@ -54,6 +51,5 @@ object DataDag:
       new DataDag(
         data,
         dag.numberOfVertices,
-        dag.adjacency,
-        dag.roots
+        dag.adjacency
       )
diff --git a/modules/core/src/main/scala/gs/graph/v0/data/DataDigraph.scala b/modules/core/src/main/scala/gs/graph/v0/data/DataDigraph.scala
index 42e3e41..f026631 100644
--- a/modules/core/src/main/scala/gs/graph/v0/data/DataDigraph.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/data/DataDigraph.scala
@@ -3,7 +3,6 @@ package gs.graph.v0.data
 import gs.graph.v0.Adjacency
 import gs.graph.v0.GraphException
 import gs.graph.v0.Size
-import gs.graph.v0.Vertex
 import gs.graph.v0.directed.Digraph
 
 /** Specialization of [[Digraph]] that associates _data_ with _each [[Vertex]]_.
@@ -11,9 +10,8 @@ import gs.graph.v0.directed.Digraph
 final class DataDigraph[A] private (
   val data: Vector[A],
   n: Size,
-  a: Adjacency,
-  r: Vector[Vertex]
-) extends Digraph(n, a, r)
+  a: Adjacency
+) extends Digraph(n, a)
       with AnyGraphWithData[A]
 
 object DataDigraph:
@@ -25,8 +23,7 @@ object DataDigraph:
     new DataDigraph[A](
       Vector.empty,
       Size.Zero,
-      Adjacency.Empty,
-      Vector.empty
+      Adjacency.Empty
     )
 
   /** Instantiate a new [[DataDigraph]] given an input [[Digraph]] and some
@@ -55,6 +52,5 @@ object DataDigraph:
       new DataDigraph(
         data,
         digraph.numberOfVertices,
-        digraph.adjacency,
-        digraph.roots
+        digraph.adjacency
       )
diff --git a/modules/core/src/main/scala/gs/graph/v0/directed/Dag.scala b/modules/core/src/main/scala/gs/graph/v0/directed/Dag.scala
index a2892f6..33a7df1 100644
--- a/modules/core/src/main/scala/gs/graph/v0/directed/Dag.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/directed/Dag.scala
@@ -3,7 +3,6 @@ package gs.graph.v0.directed
 import gs.graph.v0.Adjacency
 import gs.graph.v0.GraphError
 import gs.graph.v0.Size
-import gs.graph.v0.Vertex
 
 /** Directed Acyclic Graph (DAG): Represents a [[Digraph]] with no cycles.
   *
@@ -12,21 +11,20 @@ import gs.graph.v0.Vertex
   */
 class Dag protected (
   n: Size,
-  a: Adjacency,
-  r: Vector[Vertex]
-) extends Digraph(n, a, r)
+  a: Adjacency
+) extends Digraph(n, a)
 
 object Dag:
 
   /** The empty DAG.
     */
   final val Empty: Dag =
-    new Dag(Size.Zero, Adjacency.Empty, Vector.empty)
+    new Dag(Size.Zero, Adjacency.Empty)
 
   /** DAG with a single vertex.
     */
   final val Single: Dag =
-    new Dag(Size.One, Adjacency.Single, Vector(Vertex.Zero))
+    new Dag(Size.One, Adjacency.Single)
 
   given CanEqual[Dag, Dag]     = CanEqual.derived
   given CanEqual[Dag, Digraph] = CanEqual.derived
@@ -42,7 +40,7 @@ object Dag:
   def validate(digraph: Digraph): Either[GraphError, Dag] =
     if !Digraph.hasCycle(digraph) then
       Right(
-        new Dag(digraph.numberOfVertices, digraph.adjacency, digraph.roots)
+        new Dag(digraph.numberOfVertices, digraph.adjacency)
       )
     else Left(GraphError.CycleFound)
 
diff --git a/modules/core/src/main/scala/gs/graph/v0/directed/Digraph.scala b/modules/core/src/main/scala/gs/graph/v0/directed/Digraph.scala
index 9e3bd40..7fa4216 100644
--- a/modules/core/src/main/scala/gs/graph/v0/directed/Digraph.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/directed/Digraph.scala
@@ -13,22 +13,15 @@ import gs.graph.v0.Vertex
   *   The number of [[Vertex]] present in this graph.
   * @param adjacency
   *   The [[Adjacency]] that describes this graph.
-  * @param roots
-  *   Complete collection of root vertices.
   */
 class Digraph(
   val numberOfVertices: Size,
-  val adjacency: Adjacency,
-  val roots: Vector[Vertex]
+  val adjacency: Adjacency
 ) extends Graph:
   /** @inheritDocs
     */
   override val disposition: GraphDisposition = GraphDisposition.Directed
 
-  /** @inheritDocs
-    */
-  override def selectRoots(): Vector[Vertex] = roots
-
   /** @inheritDocs
     */
   override def equals(that: Any): Boolean =
@@ -52,13 +45,13 @@ object Digraph:
     *   An empty [[Digraph]].
     */
   final val Empty: Digraph =
-    new Digraph(Size.Zero, Adjacency.Empty, Vector.empty)
+    new Digraph(Size.Zero, Adjacency.Empty)
 
   /** @return
     *   A [[Digraph]] with one vertex.
     */
   final val Single: Digraph =
-    new Digraph(Size.One, Adjacency.Single, Vector(Vertex.Zero))
+    new Digraph(Size.One, Adjacency.Single)
 
   def fromEdges(
     numberOfVertices: Size,
@@ -66,8 +59,7 @@ object Digraph:
   ): Digraph =
     new Digraph(
       numberOfVertices = numberOfVertices,
-      adjacency = Adjacency.fromEdges(numberOfVertices, edges),
-      roots = findRootsForDirectedEdges(numberOfVertices, edges)
+      adjacency = Adjacency.fromEdges(numberOfVertices, edges)
     )
 
   /** Instantiate a new [[Graph]] from the given [[Adjacency]].
@@ -82,37 +74,9 @@ object Digraph:
   ): Digraph =
     new Digraph(
       numberOfVertices = adjacency.numberOfVertices,
-      adjacency = adjacency,
-      roots = adjacency.roots
+      adjacency = adjacency
     )
 
-  /** Find all roots for the given collection of [[Edge]].
-    *
-    * This function assumes directed edges.
-    *
-    * @param numberOfVertices
-    *   The number of vertices in the graph represented by the given edges.
-    * @param edges
-    *   The collection of edges to analyze.
-    * @return
-    *   The vector of roots for the given edges.
-    */
-  def findRootsForDirectedEdges(
-    numberOfVertices: Size,
-    edges: Iterable[Edge]
-  ): Vector[Vertex] =
-    // Count of incoming relationships.
-    val counts = Array.fill(numberOfVertices.value)(0)
-    edges.foreach(edge => counts(edge.to.ordinal) += 1)
-
-    // If there are any vertices with no incoming connections, those are roots.
-    // Note that these may be completely disconnected.
-    counts.zipWithIndex
-      .filter(_._1 == 0)
-      .map { case (_, index) => Vertex(index) }
-      .toVector
-      .distinct
-
   /** Determine whether the given [[Digraph]] has any cycles.
     *
     * See: [[Dag]]
diff --git a/modules/core/src/main/scala/gs/graph/v0/directed/SingleRootDigraph.scala b/modules/core/src/main/scala/gs/graph/v0/directed/SingleRootDigraph.scala
deleted file mode 100644
index 111a3f6..0000000
--- a/modules/core/src/main/scala/gs/graph/v0/directed/SingleRootDigraph.scala
+++ /dev/null
@@ -1,144 +0,0 @@
-package gs.graph.v0.directed
-
-import gs.graph.v0.Adjacency
-import gs.graph.v0.Edge
-import gs.graph.v0.GraphException
-import gs.graph.v0.Size
-import gs.graph.v0.Vertex
-
-/** Specialization of [[Digraph]] that always has a single root vertex.
-  *
-  * @param n
-  *   The number of [[Vertex]] present in this graph.
-  * @param a
-  *   The [[Adjacency]] that describes this graph.
-  * @param r
-  *   The singular root [[Vertex]].
-  */
-class SingleRootDigraph(
-  n: Size,
-  a: Adjacency,
-  root: Vertex
-) extends Digraph(n, a, Vector(root))
-
-object SingleRootDigraph:
-
-  /** Attempt to show that the given [[Digraph]] has a single root.
-    *
-    * @param dg
-    *   The input [[Digraph]].
-    * @return
-    *   [[SingleRootDigraph]] or `None` if the number of roots is not 1.
-    */
-  def fromDirectedGraph(dg: Digraph): Option[SingleRootDigraph] =
-    if dg.roots.size == 1 then
-      Some(
-        new SingleRootDigraph(
-          dg.numberOfVertices,
-          dg.adjacency,
-          dg.roots(0)
-        )
-      )
-    else None
-
-  /** Given some edges, build a [[SingleRootDigraph]]. Throw an exception if
-    * this operation fails.
-    *
-    * @param numberOfVertices
-    *   The number of [[Vertex]] in the graph.
-    * @param edges
-    *   The collection of [[Edge]] that describe the graph.
-    * @param root
-    *   The root [[Vertex]].
-    * @return
-    *   The new [[SingleRootDigraph]].
-    */
-  def fromEdgesUnsafe(
-    numberOfVertices: Size,
-    edges: Iterable[Edge],
-    root: Vertex
-  ): SingleRootDigraph =
-    if root < numberOfVertices then
-      new SingleRootDigraph(
-        numberOfVertices,
-        Adjacency.fromEdges(numberOfVertices, edges),
-        root
-      )
-    else throw GraphException.RootOutOfBounds(root, numberOfVertices)
-
-  /** Given some edges, build a [[SingleRootDigraph]] if that collection
-    * represents a graph with a single root.
-    *
-    * @param numberOfVertices
-    *   The number of [[Vertex]] in the graph.
-    * @param edges
-    *   The collection of [[Edge]] that describe the graph.
-    * @return
-    *   The new [[SingleRootDigraph]], or `None` if the edges do not describe a
-    *   digraph with a single root.
-    */
-  def fromEdges(
-    numberOfVertices: Size,
-    edges: Iterable[Edge]
-  ): Option[SingleRootDigraph] =
-    val roots = Digraph.findRootsForDirectedEdges(numberOfVertices, edges)
-    if roots.size == 1 then
-      Some(
-        new SingleRootDigraph(
-          numberOfVertices,
-          Adjacency.fromEdges(numberOfVertices, edges),
-          roots(0)
-        )
-      )
-    else None
-
-  /** Given some [[Adjacency]] and a given root [[Vertex]], instantiate a new
-    * [[SingleRootDigraph]].
-    *
-    * Throws an exception if the given root is not contained within the
-    * [[Adjacency]].
-    *
-    * @param adjacency
-    *   The [[Adjacency]] which describes the graph.
-    * @param root
-    *   The root [[Vertex]].
-    * @return
-    *   New [[SingleRootDigraph]]
-    */
-  def fromAdjacencyUnsafe(
-    adjacency: Adjacency,
-    root: Vertex
-  ): SingleRootDigraph =
-    if root < adjacency.numberOfVertices then
-      new SingleRootDigraph(
-        adjacency.numberOfVertices,
-        adjacency,
-        root
-      )
-    else throw GraphException.RootOutOfBounds(root, adjacency.numberOfVertices)
-
-  /** Given some [[Adjacency]] and a given root [[Vertex]], instantiate a new
-    * [[SingleRootDigraph]] if the [[Vertex]] is within the graph..
-    *
-    * @param adjacency
-    *   The [[Adjacency]] which describes the graph.
-    * @param root
-    *   The root [[Vertex]].
-    * @return
-    *   New [[SingleRootDigraph]], or `None` if the given root is not valid.
-    */
-  def fromAdjacency(
-    adjacency: Adjacency
-  ): Option[SingleRootDigraph] =
-    val roots = adjacency.roots
-    if roots.size == 1 then
-      Some(
-        new SingleRootDigraph(
-          adjacency.numberOfVertices,
-          adjacency,
-          roots(0)
-        )
-      )
-    else None
-
-end SingleRootDigraph
diff --git a/modules/core/src/main/scala/gs/graph/v0/directed/StronglyConnectedComponent.scala b/modules/core/src/main/scala/gs/graph/v0/directed/StronglyConnectedComponent.scala
index 37b28f8..2e9dc28 100644
--- a/modules/core/src/main/scala/gs/graph/v0/directed/StronglyConnectedComponent.scala
+++ b/modules/core/src/main/scala/gs/graph/v0/directed/StronglyConnectedComponent.scala
@@ -1,5 +1,8 @@
 package gs.graph.v0.directed
 
+import gs.graph.v0.Adjacency
+import gs.graph.v0.Edge
+import gs.graph.v0.MappedAdjacency
 import gs.graph.v0.Vertex
 import scala.collection.mutable.ListBuffer
 import scala.collection.mutable.Stack
@@ -10,49 +13,50 @@ import scala.collection.mutable.Stack
   *
   * This class is used with [[Digraph]]. See [[gs.graph.v0.ConnectedComponent]]
   * for general connected components.
-  *
-  * TODO: ADD EDGES
   */
-final class StronglyConnectedComponent private (val value: Set[Vertex]):
+final class StronglyConnectedComponent private (
+  val vertices: Set[Vertex],
+  val adjacency: Adjacency
+):
 
   /** @inheritDocs
     */
   override def equals(obj: Any): Boolean =
     obj match
-      case other: StronglyConnectedComponent => value.equals(other.value)
-      case _                                 => false
+      case other: StronglyConnectedComponent =>
+        vertices.equals(other.vertices) && adjacency.equals(other.adjacency)
+      case _ => false
 
   /** @inheritDocs
     */
-  override def hashCode(): Int = value.hashCode()
+  override def hashCode(): Int = adjacency.hashCode()
 
   /** @inheritDocs
     */
   override def toString(): String =
     val sb = StringBuilder()
-    sb.append("[")
-    sb.append(value.mkString(","))
-    sb.append("]")
+    sb.append("entries:[\n")
+    sb.append(
+      adjacency.entries
+        .map { case (v, ns) => s"$v -> ${ns.mkString(",")}" }
+        .mkString("\n")
+    )
+    sb.append("\n]")
     sb.toString()
 
   /** @return
     *   The number of vertices contained by this SCC.
     */
-  def size: Int = value.size
+  lazy val size: Int = vertices.size
 
   /** @return
     *   True if this SCC contains a single vertex.
     */
-  def isSingle: Boolean = value.size == 1
-
-  /** @return
-    *   The set of vertices in the SCC.
-    */
-  def vertices: Set[Vertex] = value
+  def isSingle: Boolean = size == 1
 
   /** Retrieve an arbitrary SCC member as the root vertex.
     */
-  lazy val root: Vertex = value.toList.head
+  lazy val root: Vertex = vertices.toList.head
 
 end StronglyConnectedComponent
 
@@ -67,19 +71,29 @@ object StronglyConnectedComponent:
     * @return
     *   The new SCC.
     */
-  def apply(value: Vertex*): StronglyConnectedComponent =
-    if value.isEmpty then
+  def apply(connections: Edge*): StronglyConnectedComponent =
+    if connections.isEmpty then
       throw new IllegalArgumentException(
         "Empty strongly connected components (SCC) are not allowed."
       )
-    else new StronglyConnectedComponent(value.toSet)
+    else
+      val adj = MappedAdjacency.fromEdges(connections)
+      new StronglyConnectedComponent(
+        vertices = adj.vertices.toSet,
+        adjacency = adj
+      )
 
-  def of(value: Int*): StronglyConnectedComponent =
+  def of(value: (Int, Int)*): StronglyConnectedComponent =
     if value.isEmpty then
       throw new IllegalArgumentException(
         "Empty strongly connected components (SCC) are not allowed."
       )
-    else new StronglyConnectedComponent(value.map(Vertex.apply).toSet)
+    else
+      val adj = MappedAdjacency.fromEdges(value.map(Edge.fromPair))
+      new StronglyConnectedComponent(
+        vertices = adj.vertices.toSet,
+        adjacency = adj
+      )
 
   /** Instantiate a new SCC that contains a single [[Vertex]].
     *
@@ -89,7 +103,10 @@ object StronglyConnectedComponent:
     *   The new SCC.
     */
   def single(value: Vertex): StronglyConnectedComponent =
-    new StronglyConnectedComponent(Set(value))
+    new StronglyConnectedComponent(
+      vertices = Set(value),
+      adjacency = MappedAdjacency.single(value)
+    )
 
   given CanEqual[StronglyConnectedComponent, StronglyConnectedComponent] =
     CanEqual.derived
@@ -201,9 +218,17 @@ object StronglyConnectedComponent:
         sccVertex = s.pop()
 
       scc.addOne(sccVertex)
-      if !scc.isEmpty then
-        output.addOne(new StronglyConnectedComponent(scc.toSet))
-      else println("ERROR")
+
+      // These accumulated vertices make up the SCC.
+      val vertices = scc.toSet
+
+      output.addOne(
+        new StronglyConnectedComponent(
+          vertices = scc.toSet,
+          // Select the subset of the adjacency which is relevant to the SCC.
+          adjacency = g.adjacency.subset(vertices)
+        )
+      )
     else ()
 
   /** Mutable state wrapper around an integer.
diff --git a/modules/core/src/test/scala/gs/graph/v0/UndirectedGraphTests.scala b/modules/core/src/test/scala/gs/graph/v0/UndirectedGraphTests.scala
index def71cc..635a51d 100644
--- a/modules/core/src/test/scala/gs/graph/v0/UndirectedGraphTests.scala
+++ b/modules/core/src/test/scala/gs/graph/v0/UndirectedGraphTests.scala
@@ -9,7 +9,6 @@ class UndirectedGraphTests extends FunSuite:
     assertEquals(g.numberOfVertices, Size.Zero)
     assertEquals(g.adjacency, Adjacency.Empty)
     assertEquals(g.disposition, GraphDisposition.Undirected)
-    assertEquals(g.selectRoots(), Vector.empty)
   }
 
   test("should construct a graph based on # of edges and edge list") {
@@ -24,7 +23,6 @@ class UndirectedGraphTests extends FunSuite:
     assertEquals(g.numberOfVertices, N)
     assertEquals(g.neighbors(Vertex(0)), Vector(Vertex(1), Vertex(2)))
     assertEquals(g.neighbors(Vertex(4)), Vector.empty)
-    assertEquals(g.selectRoots(), Vector(Vertex.Zero))
   }
 
   test("should construct a graph based on edge list") {
@@ -38,7 +36,6 @@ class UndirectedGraphTests extends FunSuite:
     assertEquals(g.numberOfVertices, N)
     assertEquals(g.neighbors(Vertex(0)), Vector(Vertex(1), Vertex(2)))
     assertEquals(g.neighbors(Vertex(4)), Vector.empty)
-    assertEquals(g.selectRoots(), Vector(Vertex.Zero))
   }
 
   test("should calculate roots for each connected component") {