Class AsSynchronizedGraph<V,E>

java.lang.Object
org.jgrapht.graph.AbstractGraph<V,E>
org.jgrapht.graph.GraphDelegator<V,E>
org.jgrapht.graph.concurrent.AsSynchronizedGraph<V,E>
Type Parameters:
V - the graph vertex type
E - the graph edge type
All Implemented Interfaces:
Serializable, Graph<V,E>

public class AsSynchronizedGraph<V,E> extends GraphDelegator<V,E> implements Graph<V,E>, Serializable
Create a synchronized (thread-safe) Graph backed by the specified Graph. This Graph is designed to support concurrent reads which are mutually exclusive with writes. In order to guarantee serial access, it is critical that all access to the backing Graph is accomplished through the created Graph.

Users need to manually synchronize on edge supplier (see Graph.getEdgeSupplier()) if creating an edge needs to access shared resources. Failure to follow this advice may result in non-deterministic behavior.

For all methods returning a Set, the Graph guarantees that all operations on the returned Set do not affect the backing Graph. For edgeSet and vertexSet methods, the returned Set is backed by the underlying graph, but when a traversal over the set is started via a method such as iterator(), a snapshot of the underlying Set is copied for iteration purposes. For edgesOf, incomingEdgesOf and outgoingEdgesOf methods, the returned Set is a unmodifiable copy of the result produced by the underlying Graph. Users can control whether those copies should be cached; caching may significantly increase memory requirements. If users decide to cache those copies and the backing graph's changes don't affect them, those copies will be returned the next time the method is called. If the backing graph's changes affect them, they will be removed from cache and re-created the next time the method is called. If users decide to not cache those copies, the graph will create ephemeral copies every time the method is called. For other methods returning a Set, the Set is just the backing Graph's return.

As an alternative, a copyless mode is supported. When enabled, no collection copies are made at all (and hence the cache setting is ignored). This requires the caller to explicitly synchronize iteration via the getLock() method. This approach requires quite a bit of care on the part of the calling application, so it is disabled by default.

Even though this graph implementation is thread-safe, callers should still be aware of potential hazards from removal methods. If calling code obtains a reference to a vertex or edge from the graph, and then calls another graph method to access information about that object, an IllegalArgumentException may be thrown if another thread has concurrently removed that object. Therefore, calling the remove methods concurrently with a typical algorithm is likely to cause the algorithm to fail with an IllegalArgumentException. So really the main concurrent read/write use case is add-only.
eg: If threadA tries to get all edges touching a certain vertex after threadB removes the vertex, the algorithm will be interrupted by IllegalArgumentException.

 Thread threadA = new Thread(() -> {
     Set vertices = graph.vertexSet();
     for (Object v : vertices) {
         // IllegalArgumentException may be thrown since other threads may have removed
         // the vertex.
         Set edges = graph.edgesOf(v);
         doOtherThings();
     }
 });
 Thread threadB = new Thread(() -> {
     Set vertices = graph.vertexSet();
     for (Object v : vertices) {
         if (someConditions) {
             graph.removeVertex(v);
         }
     }
 });
 

One way to avoid the hazard noted above is for the calling application to explicitly synchronize all iterations using the getLock() method.

The created Graph's hashCode is equal to the backing set's hashCode. And the created Graph is equal to another Graph if they are the same Graph or the backing Graph is equal to the other Graph.

Author:
CHEN Kui
See Also:
  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static class 
    A builder for AsSynchronizedGraph.
  • Field Summary

    Fields inherited from interface org.jgrapht.Graph

    DEFAULT_EDGE_WEIGHT
  • Constructor Summary

    Constructors
    Constructor
    Description
    Constructor for AsSynchronizedGraph with default settings (cache disabled, non-fair mode, and copyless mode disabled).
  • Method Summary

    Modifier and Type
    Method
    Description
    addEdge(V sourceVertex, V targetVertex)
    Creates a new edge in this graph, going from the source vertex to the target vertex, and returns the created edge.
    boolean
    addEdge(V sourceVertex, V targetVertex, E e)
    Adds the specified edge to this graph, going from the source vertex to the target vertex.
    boolean
    Adds the specified vertex to this graph if not already present.
    boolean
    Returns true if this graph contains the specified edge.
    boolean
    containsEdge(V sourceVertex, V targetVertex)
    Returns true if and only if this graph contains an edge going from the source vertex to the target vertex.
    boolean
    Returns true if this graph contains the specified vertex.
    int
    degreeOf(V vertex)
    Returns the degree of the specified vertex.
    Returns a set of the edges contained in this graph.
    edgesOf(V vertex)
    Returns a set of all edges touching the specified vertex.
    boolean
    Indicates whether some other object is "equal to" this graph.
    getAllEdges(V sourceVertex, V targetVertex)
    Returns a set of all edges connecting source vertex to target vertex if such vertices exist in this graph.
    getEdge(V sourceVertex, V targetVertex)
    Returns an edge connecting source vertex to target vertex if such vertices and such edge exist in this graph.
    Returns the source vertex of an edge.
    Returns the target vertex of an edge.
    double
    Returns the weight assigned to a given edge.
    Get the read/write lock used to synchronize all access to this graph.
    int
    Returns a hash code value for this graph.
    Returns a set of all edges incoming into the specified vertex.
    int
    inDegreeOf(V vertex)
    Returns the "in degree" of the specified vertex.
    boolean
    Return whether the graph uses cache for edgesOf, incomingEdgesOf and outgoingEdgesOf methods.
    boolean
    Return whether copyless mode is used for collection-returning methods.
    boolean
    Return whether fair mode is used for synchronizing access to this graph.
    int
    outDegreeOf(V vertex)
    Returns the "out degree" of the specified vertex.
    Returns a set of all edges outgoing from the specified vertex.
    boolean
    removeAllEdges(Collection<? extends E> edges)
    Removes all the edges in this graph that are also contained in the specified edge collection.
    removeAllEdges(V sourceVertex, V targetVertex)
    Removes all the edges going from the specified source vertex to the specified target vertex, and returns a set of all removed edges.
    boolean
    removeAllVertices(Collection<? extends V> vertices)
    Removes all the vertices in this graph that are also contained in the specified vertex collection.
    boolean
    Removes the specified edge from the graph.
    removeEdge(V sourceVertex, V targetVertex)
    Removes an edge going from source vertex to target vertex, if such vertices and such edge exist in this graph.
    boolean
    Removes the specified vertex from this graph including all its touching edges if present.
    setCache(boolean cacheEnabled)
    Set the cache strategy for edgesOf, incomingEdgesOf and outgoingEdgesOf methods.
    void
    setEdgeWeight(E e, double weight)
    Assigns a weight to an edge.
    Returns a string of the parenthesized pair (V, E) representing this G=(V,E) graph.
    Returns a set of the vertices contained in this graph.

    Methods inherited from class org.jgrapht.graph.GraphDelegator

    addVertex, getDelegate, getEdgeSupplier, getType, getVertexSupplier

    Methods inherited from class org.jgrapht.graph.AbstractGraph

    assertVertexExist, removeAllEdges, toStringFromSets

    Methods inherited from class java.lang.Object

    clone, finalize, getClass, notify, notifyAll, wait, wait, wait

    Methods inherited from interface org.jgrapht.Graph

    addVertex, getEdgeSupplier, getType, getVertexSupplier, iterables, setEdgeWeight
  • Constructor Details

    • AsSynchronizedGraph

      public AsSynchronizedGraph(Graph<V,E> g)
      Constructor for AsSynchronizedGraph with default settings (cache disabled, non-fair mode, and copyless mode disabled).
      Parameters:
      g - the backing graph (the delegate)
  • Method Details

    • getAllEdges

      public Set<E> getAllEdges(V sourceVertex, V targetVertex)
      Returns a set of all edges connecting source vertex to target vertex if such vertices exist in this graph. If any of the vertices does not exist or is null, returns null. If both vertices exist but no edges found, returns an empty set.

      In undirected graphs, some of the returned edges may have their source and target vertices in the opposite order. In simple graphs the returned set is either singleton set or empty set.

      Specified by:
      getAllEdges in interface Graph<V,E>
      Overrides:
      getAllEdges in class GraphDelegator<V,E>
      Parameters:
      sourceVertex - source vertex of the edge.
      targetVertex - target vertex of the edge.
      Returns:
      a set of all edges connecting source vertex to target vertex.
    • getEdge

      public E getEdge(V sourceVertex, V targetVertex)
      Returns an edge connecting source vertex to target vertex if such vertices and such edge exist in this graph. Otherwise returns null. If any of the specified vertices is null returns null

      In undirected graphs, the returned edge may have its source and target vertices in the opposite order.

      Specified by:
      getEdge in interface Graph<V,E>
      Overrides:
      getEdge in class GraphDelegator<V,E>
      Parameters:
      sourceVertex - source vertex of the edge.
      targetVertex - target vertex of the edge.
      Returns:
      an edge connecting source vertex to target vertex.
    • addEdge

      public E addEdge(V sourceVertex, V targetVertex)
      Creates a new edge in this graph, going from the source vertex to the target vertex, and returns the created edge. Some graphs do not allow edge-multiplicity. In such cases, if the graph already contains an edge from the specified source to the specified target, then this method does not change the graph and returns null.

      The source and target vertices must already be contained in this graph. If they are not found in graph IllegalArgumentException is thrown.

      This method creates the new edge e using this graph's edge supplier (see Graph.getEdgeSupplier()). For the new edge to be added e must not be equal to any other edge the graph (even if the graph allows edge-multiplicity). More formally, the graph must not contain any edge e2 such that e2.equals(e). If such e2 is found then the newly created edge e is abandoned, the method leaves this graph unchanged and returns null.

      If the underlying graph implementation's Graph.getEdgeSupplier() returns null, then this method cannot create edges and throws an UnsupportedOperationException.

      Specified by:
      addEdge in interface Graph<V,E>
      Overrides:
      addEdge in class GraphDelegator<V,E>
      Parameters:
      sourceVertex - source vertex of the edge.
      targetVertex - target vertex of the edge.
      Returns:
      The newly created edge if added to the graph, otherwise null.
      See Also:
    • addEdge

      public boolean addEdge(V sourceVertex, V targetVertex, E e)
      Adds the specified edge to this graph, going from the source vertex to the target vertex. More formally, adds the specified edge, e, to this graph if this graph contains no edge e2 such that e2.equals(e). If this graph already contains such an edge, the call leaves this graph unchanged and returns false. Some graphs do not allow edge-multiplicity. In such cases, if the graph already contains an edge from the specified source to the specified target, then this method does not change the graph and returns false. If the edge was added to the graph, returns true.

      The source and target vertices must already be contained in this graph. If they are not found in graph IllegalArgumentException is thrown.

      Specified by:
      addEdge in interface Graph<V,E>
      Overrides:
      addEdge in class GraphDelegator<V,E>
      Parameters:
      sourceVertex - source vertex of the edge.
      targetVertex - target vertex of the edge.
      e - edge to be added to this graph.
      Returns:
      true if this graph did not already contain the specified edge.
      See Also:
    • addVertex

      public boolean addVertex(V v)
      Adds the specified vertex to this graph if not already present. More formally, adds the specified vertex, v, to this graph if this graph contains no vertex u such that u.equals(v). If this graph already contains such vertex, the call leaves this graph unchanged and returns false. In combination with the restriction on constructors, this ensures that graphs never contain duplicate vertices.
      Specified by:
      addVertex in interface Graph<V,E>
      Overrides:
      addVertex in class GraphDelegator<V,E>
      Parameters:
      v - vertex to be added to this graph.
      Returns:
      true if this graph did not already contain the specified vertex.
    • containsEdge

      public boolean containsEdge(V sourceVertex, V targetVertex)
      Returns true if and only if this graph contains an edge going from the source vertex to the target vertex. In undirected graphs the same result is obtained when source and target are inverted. If any of the specified vertices does not exist in the graph, or if is null, returns false.
      Specified by:
      containsEdge in interface Graph<V,E>
      Overrides:
      containsEdge in class AbstractGraph<V,E>
      Parameters:
      sourceVertex - source vertex of the edge.
      targetVertex - target vertex of the edge.
      Returns:
      true if this graph contains the specified edge.
      See Also:
    • containsEdge

      public boolean containsEdge(E e)
      Returns true if this graph contains the specified edge. More formally, returns true if and only if this graph contains an edge e2 such that e.equals(e2). If the specified edge is null returns false.
      Specified by:
      containsEdge in interface Graph<V,E>
      Overrides:
      containsEdge in class GraphDelegator<V,E>
      Parameters:
      e - edge whose presence in this graph is to be tested.
      Returns:
      true if this graph contains the specified edge.
    • containsVertex

      public boolean containsVertex(V v)
      Returns true if this graph contains the specified vertex. More formally, returns true if and only if this graph contains a vertex u such that u.equals(v). If the specified vertex is null returns false.
      Specified by:
      containsVertex in interface Graph<V,E>
      Overrides:
      containsVertex in class GraphDelegator<V,E>
      Parameters:
      v - vertex whose presence in this graph is to be tested.
      Returns:
      true if this graph contains the specified vertex.
    • degreeOf

      public int degreeOf(V vertex)
      Returns the degree of the specified vertex.
      Specified by:
      degreeOf in interface Graph<V,E>
      Overrides:
      degreeOf in class GraphDelegator<V,E>
      Parameters:
      vertex - vertex whose degree is to be calculated
      Returns:
      the degree of the specified vertex
    • edgeSet

      public Set<E> edgeSet()
      Returns a set of the edges contained in this graph. The set is backed by the graph, so changes to the graph are reflected in the set. If the graph is modified while an iteration over the set is in progress, the results of the iteration are undefined.

      The graph implementation may maintain a particular set ordering (e.g. via LinkedHashSet) for deterministic iteration, but this is not required. It is the responsibility of callers who rely on this behavior to only use graph implementations which support it.

      Specified by:
      edgeSet in interface Graph<V,E>
      Overrides:
      edgeSet in class GraphDelegator<V,E>
      Returns:
      a set of the edges contained in this graph.
    • edgesOf

      public Set<E> edgesOf(V vertex)
      Returns a set of all edges touching the specified vertex. If no edges are touching the specified vertex returns an empty set.
      Specified by:
      edgesOf in interface Graph<V,E>
      Overrides:
      edgesOf in class GraphDelegator<V,E>
      Parameters:
      vertex - the vertex for which a set of touching edges is to be returned.
      Returns:
      a set of all edges touching the specified vertex.
    • inDegreeOf

      public int inDegreeOf(V vertex)
      Returns the "in degree" of the specified vertex.

      The "in degree" of a vertex in a directed graph is the number of inward directed edges from that vertex. See http://mathworld.wolfram.com/Indegree.html.

      In the case of undirected graphs this method returns the number of edges touching the vertex. Edges with same source and target vertices (self-loops) are counted twice.

      Specified by:
      inDegreeOf in interface Graph<V,E>
      Overrides:
      inDegreeOf in class GraphDelegator<V,E>
      Parameters:
      vertex - vertex whose degree is to be calculated.
      Returns:
      the degree of the specified vertex.
    • incomingEdgesOf

      public Set<E> incomingEdgesOf(V vertex)
      Returns a set of all edges incoming into the specified vertex.

      In the case of undirected graphs this method returns all edges touching the vertex, thus, some of the returned edges may have their source and target vertices in the opposite order.

      Specified by:
      incomingEdgesOf in interface Graph<V,E>
      Overrides:
      incomingEdgesOf in class GraphDelegator<V,E>
      Parameters:
      vertex - the vertex for which the list of incoming edges to be returned.
      Returns:
      a set of all edges incoming into the specified vertex.
    • outDegreeOf

      public int outDegreeOf(V vertex)
      Returns the "out degree" of the specified vertex.

      The "out degree" of a vertex in a directed graph is the number of outward directed edges from that vertex. See http://mathworld.wolfram.com/Outdegree.html.

      In the case of undirected graphs this method returns the number of edges touching the vertex. Edges with same source and target vertices (self-loops) are counted twice.

      Specified by:
      outDegreeOf in interface Graph<V,E>
      Overrides:
      outDegreeOf in class GraphDelegator<V,E>
      Parameters:
      vertex - vertex whose degree is to be calculated.
      Returns:
      the degree of the specified vertex.
    • outgoingEdgesOf

      public Set<E> outgoingEdgesOf(V vertex)
      Returns a set of all edges outgoing from the specified vertex.

      In the case of undirected graphs this method returns all edges touching the vertex, thus, some of the returned edges may have their source and target vertices in the opposite order.

      Specified by:
      outgoingEdgesOf in interface Graph<V,E>
      Overrides:
      outgoingEdgesOf in class GraphDelegator<V,E>
      Parameters:
      vertex - the vertex for which the list of outgoing edges to be returned.
      Returns:
      a set of all edges outgoing from the specified vertex.
    • removeAllEdges

      public boolean removeAllEdges(Collection<? extends E> edges)
      Removes all the edges in this graph that are also contained in the specified edge collection. After this call returns, this graph will contain no edges in common with the specified edges. This method will invoke the Graph.removeEdge(Object) method.
      Specified by:
      removeAllEdges in interface Graph<V,E>
      Overrides:
      removeAllEdges in class AbstractGraph<V,E>
      Parameters:
      edges - edges to be removed from this graph.
      Returns:
      true if this graph changed as a result of the call
      See Also:
    • removeAllEdges

      public Set<E> removeAllEdges(V sourceVertex, V targetVertex)
      Removes all the edges going from the specified source vertex to the specified target vertex, and returns a set of all removed edges. Returns null if any of the specified vertices does not exist in the graph. If both vertices exist but no edge is found, returns an empty set. This method will either invoke the Graph.removeEdge(Object) method, or the Graph.removeEdge(Object, Object) method.
      Specified by:
      removeAllEdges in interface Graph<V,E>
      Overrides:
      removeAllEdges in class AbstractGraph<V,E>
      Parameters:
      sourceVertex - source vertex of the edge.
      targetVertex - target vertex of the edge.
      Returns:
      the removed edges, or null if either vertex is not part of graph
      See Also:
    • removeAllVertices

      public boolean removeAllVertices(Collection<? extends V> vertices)
      Removes all the vertices in this graph that are also contained in the specified vertex collection. After this call returns, this graph will contain no vertices in common with the specified vertices. This method will invoke the Graph.removeVertex(Object) method.
      Specified by:
      removeAllVertices in interface Graph<V,E>
      Overrides:
      removeAllVertices in class AbstractGraph<V,E>
      Parameters:
      vertices - vertices to be removed from this graph.
      Returns:
      true if this graph changed as a result of the call
      See Also:
    • removeEdge

      public boolean removeEdge(E e)
      Removes the specified edge from the graph. Removes the specified edge from this graph if it is present. More formally, removes an edge e2 such that e2.equals(e), if the graph contains such edge. Returns true if the graph contained the specified edge. (The graph will not contain the specified edge once the call returns).

      If the specified edge is null returns false.

      Specified by:
      removeEdge in interface Graph<V,E>
      Overrides:
      removeEdge in class GraphDelegator<V,E>
      Parameters:
      e - edge to be removed from this graph, if present.
      Returns:
      true if and only if the graph contained the specified edge.
    • removeEdge

      public E removeEdge(V sourceVertex, V targetVertex)
      Removes an edge going from source vertex to target vertex, if such vertices and such edge exist in this graph. Returns the edge if removed or null otherwise.
      Specified by:
      removeEdge in interface Graph<V,E>
      Overrides:
      removeEdge in class GraphDelegator<V,E>
      Parameters:
      sourceVertex - source vertex of the edge.
      targetVertex - target vertex of the edge.
      Returns:
      The removed edge, or null if no edge removed.
    • removeVertex

      public boolean removeVertex(V v)
      Removes the specified vertex from this graph including all its touching edges if present. More formally, if the graph contains a vertex u such that u.equals(v), the call removes all edges that touch u and then removes u itself. If no such u is found, the call leaves the graph unchanged. Returns true if the graph contained the specified vertex. (The graph will not contain the specified vertex once the call returns).

      If the specified vertex is null returns false.

      Specified by:
      removeVertex in interface Graph<V,E>
      Overrides:
      removeVertex in class GraphDelegator<V,E>
      Parameters:
      v - vertex to be removed from this graph, if present.
      Returns:
      true if the graph contained the specified vertex; false otherwise.
    • toString

      public String toString()
      Returns a string of the parenthesized pair (V, E) representing this G=(V,E) graph. 'V' is the string representation of the vertex set, and 'E' is the string representation of the edge set.
      Overrides:
      toString in class GraphDelegator<V,E>
      Returns:
      a string representation of this graph.
    • vertexSet

      public Set<V> vertexSet()
      Returns a set of the vertices contained in this graph. The set is backed by the graph, so changes to the graph are reflected in the set. If the graph is modified while an iteration over the set is in progress, the results of the iteration are undefined.

      The graph implementation may maintain a particular set ordering (e.g. via LinkedHashSet) for deterministic iteration, but this is not required. It is the responsibility of callers who rely on this behavior to only use graph implementations which support it.

      Specified by:
      vertexSet in interface Graph<V,E>
      Overrides:
      vertexSet in class GraphDelegator<V,E>
      Returns:
      a set view of the vertices contained in this graph.
    • getEdgeSource

      public V getEdgeSource(E e)
      Returns the source vertex of an edge. For an undirected graph, source and target are distinguishable designations (but without any mathematical meaning).
      Specified by:
      getEdgeSource in interface Graph<V,E>
      Overrides:
      getEdgeSource in class GraphDelegator<V,E>
      Parameters:
      e - edge of interest
      Returns:
      source vertex
    • getEdgeTarget

      public V getEdgeTarget(E e)
      Returns the target vertex of an edge. For an undirected graph, source and target are distinguishable designations (but without any mathematical meaning).
      Specified by:
      getEdgeTarget in interface Graph<V,E>
      Overrides:
      getEdgeTarget in class GraphDelegator<V,E>
      Parameters:
      e - edge of interest
      Returns:
      target vertex
    • getEdgeWeight

      public double getEdgeWeight(E e)
      Returns the weight assigned to a given edge. Unweighted graphs return 1.0 (as defined by Graph.DEFAULT_EDGE_WEIGHT), allowing weighted-graph algorithms to apply to them when meaningful.
      Specified by:
      getEdgeWeight in interface Graph<V,E>
      Overrides:
      getEdgeWeight in class GraphDelegator<V,E>
      Parameters:
      e - edge of interest
      Returns:
      edge weight
    • setEdgeWeight

      public void setEdgeWeight(E e, double weight)
      Assigns a weight to an edge.
      Specified by:
      setEdgeWeight in interface Graph<V,E>
      Overrides:
      setEdgeWeight in class GraphDelegator<V,E>
      Parameters:
      e - edge on which to set weight
      weight - new weight for edge
    • isCacheEnabled

      public boolean isCacheEnabled()
      Return whether the graph uses cache for edgesOf, incomingEdgesOf and outgoingEdgesOf methods.
      Returns:
      true if cache is in use, false if cache is not in use.
    • isCopyless

      public boolean isCopyless()
      Return whether copyless mode is used for collection-returning methods.
      Returns:
      true if the graph uses copyless mode, false otherwise
    • setCache

      public AsSynchronizedGraph<V,E> setCache(boolean cacheEnabled)
      Set the cache strategy for edgesOf, incomingEdgesOf and outgoingEdgesOf methods.
      Parameters:
      cacheEnabled - a flag whether to use cache for those methods, if true, cache will be used for those methods, otherwise cache will not be used.
      Returns:
      the AsSynchronizedGraph
    • hashCode

      public int hashCode()
      Returns a hash code value for this graph. The hash code of a graph is defined to be the sum of the hash codes of vertices and edges in the graph. It is also based on graph topology and edges weights.
      Overrides:
      hashCode in class AbstractGraph<V,E>
      Returns:
      the hash code value this graph
      See Also:
    • equals

      public boolean equals(Object o)
      Indicates whether some other object is "equal to" this graph. Returns true if the given object is also a graph, the two graphs are instances of the same graph class, have identical vertices and edges sets with the same weights.
      Overrides:
      equals in class AbstractGraph<V,E>
      Parameters:
      o - object to be compared for equality with this graph
      Returns:
      true if the specified object is equal to this graph
      See Also:
    • isFair

      public boolean isFair()
      Return whether fair mode is used for synchronizing access to this graph.
      Returns:
      true if the graph uses fair mode, false if non-fair mode
    • getLock

      public ReentrantReadWriteLock getLock()
      Get the read/write lock used to synchronize all access to this graph. This can be used by calling applications to explicitly synchronize compound sequences of graph accessses. The lock is reentrant, so the locks acquired internally by AsSynchronizedGraph will not interfere with the caller's acquired lock. However, write methods MUST NOT be called while holding a read lock, otherwise a deadlock will occur.
      Returns:
      the reentrant read/write lock used to synchronize all access to this graph