# graph - Man Page

Create and manipulate directed graph objects

## Synopsis

`package require `

**Tcl 8.4**

`package require `

**struct::graph ?2.4.1?**

`package require `

**struct::list ?1.5?**

`package require `

**struct::set ?2.2.3?**

**::struct::graph** ?*graphName*? ?**=**|**:=**|**as**|**deserialize** *source*?

**graphName** *option* ?*arg arg ...*?

*graphName* **=** *sourcegraph*

*graphName* **-->** *destgraph*

*graphName* **append** *key value*

*graphName* **deserialize** *serialization*

*graphName* **destroy**

*graphName* **arc append** *arc key value*

*graphName* **arc attr** *key*

*graphName* **arc attr** *key* **-arcs** *list*

*graphName* **arc attr** *key* **-glob** *globpattern*

*graphName* **arc attr** *key* **-regexp** *repattern*

*graphName* **arc delete** *arc* ?*arc* ...?

*graphName* **arc exists** *arc*

*graphName* **arc flip** *arc*

*graphName* **arc get** *arc key*

*graphName* **arc getall** *arc* ?*pattern*?

*graphName* **arc getunweighted**

*graphName* **arc getweight** *arc*

*graphName* **arc keys** *arc* ?*pattern*?

*graphName* **arc keyexists** *arc key*

*graphName* **arc insert** *start end* ?*child*?

*graphName* **arc lappend** *arc key value*

*graphName* **arc rename** *arc newname*

*graphName* **arc set** *arc key* ?*value*?

*graphName* **arc setunweighted** ?*weight*?

*graphName* **arc setweight** *arc weight*

*graphName* **arc unsetweight** *arc*

*graphName* **arc hasweight** *arc*

*graphName* **arc source** *arc*

*graphName* **arc target** *arc*

*graphName* **arc nodes** *arc*

*graphName* **arc move-source** *arc newsource*

*graphName* **arc move-target** *arc newtarget*

*graphName* **arc move** *arc newsource newtarget*

*graphName* **arc unset** *arc key*

*graphName* **arc weights**

*graphName* **arcs** ?-key *key*? ?-value *value*? ?-filter *cmdprefix*? ?-in|-out|-adj|-inner|-embedding *node node...*?

*graphName* **lappend** *key value*

*graphName* **node append** *node key value*

*graphName* **node attr** *key*

*graphName* **node attr** *key* **-nodes** *list*

*graphName* **node attr** *key* **-glob** *globpattern*

*graphName* **node attr** *key* **-regexp** *repattern*

*graphName* **node degree** ?-in|-out? *node*

*graphName* **node delete** *node* ?*node*...?

*graphName* **node exists** *node*

*graphName* **node get** *node key*

*graphName* **node getall** *node* ?*pattern*?

*graphName* **node keys** *node* ?*pattern*?

*graphName* **node keyexists** *node key*

*graphName* **node insert** ?*node*...?

*graphName* **node lappend** *node key value*

*graphName* **node opposite** *node arc*

*graphName* **node rename** *node newname*

*graphName* **node set** *node key* ?*value*?

*graphName* **node unset** *node key*

*graphName* **nodes** ?-key *key*? ?-value *value*? ?-filter *cmdprefix*? ?-in|-out|-adj|-inner|-embedding *node node*...?

*graphName* **get** *key*

*graphName* **getall** ?*pattern*?

*graphName* **keys** ?*pattern*?

*graphName* **keyexists** *key*

*graphName* **serialize** ?*node*...?

*graphName* **set** *key* ?*value*?

*graphName* **swap** *node1 node2*

*graphName* **unset** *key*

*graphName* **walk** *node* ?-order *order*? ?-type *type*? ?-dir *direction*? -command *cmd*

## Description

A directed graph is a structure containing two collections of elements, called *nodes* and *arcs* respectively, together with a relation ("connectivity") that places a general structure upon the nodes and arcs.

Each arc is connected to two nodes, one of which is called the *source* and the other the *target*. This imposes a direction upon the arc, which is said to go from the source to the target. It is allowed that source and target of an arc are the same node. Such an arc is called a *loop*. Whenever a node is either the source or target of an arc both are said to be *adjacent*. This extends into a relation between nodes, i.e. if two nodes are connected through at least one arc they are said to be *adjacent* too.

Each node can be the source and target for any number of arcs. The former are called the *outgoing arcs* of the node, the latter the *incoming arcs* of the node. The number of arcs in either set is called the *in-degree* resp. the *out-degree* of the node.

In addition to maintaining the node and arc relationships, this graph implementation allows any number of named *attributes* to be associated with the graph itself, and each node or arc.

*Note:* The major version of the package **struct** has been changed to version 2.0, due to backward incompatible changes in the API of this module. Please read the section **Changes for 2.0** for a full list of all changes, incompatible and otherwise.

*Note:* A C-implementation of the command can be had from the location http://www.purl.org/NET/schlenker/tcl/cgraph. See also http://wiki.tcl.tk/cgraph. This implementation uses a bit less memory than the tcl version provided here directly, and is faster. Its support is limited to versions of the package before 2.0.

As of version 2.2 of this package a critcl based C implementation is available from here as well. This implementation however requires Tcl 8.4 to run.

The main command of the package is:

**::struct::graph**?*graphName*? ?**=**|**:=**|**as**|**deserialize***source*?The command creates a new graph object with an associated global Tcl command whose name is

*graphName*. This command may be used to invoke various operations on the graph. It has the following general form:**graphName***option*?*arg arg ...*?*Option*and the*arg*s determine the exact behavior of the command.

If *graphName* is not specified a unique name will be generated by the package itself. If a *source* is specified the new graph will be initialized to it. For the operators **=**, **:=**, and **as** the *source* argument is interpreted as the name of another graph object, and the assignment operator **=** will be executed. For the operator **deserialize** the *source* is a serialized graph object and **deserialize** will be executed.

In other words

::struct::graph mygraph = b

is equivalent to

::struct::graph mygraph mygraph = b

and

::struct::graph mygraph deserialize $b

is equivalent to

::struct::graph mygraph mygraph deserialize $b

The following commands are possible for graph objects:

*graphName***=***sourcegraph*This is the

*assignment*operator for graph objects. It copies the graph contained in the graph object*sourcegraph*over the graph data in*graphName*. The old contents of*graphName*are deleted by this operation.This operation is in effect equivalent to

*graphName***deserialize**[*sourcegraph***serialize**]

The operation assumes that the *sourcegraph* provides the method **serialize** and that this method returns a valid graph serialization.

*graphName***-->***destgraph*This is the

*reverse assignment*operator for graph objects. It copies the graph contained in the graph object*graphName*over the graph data in the object*destgraph*. The old contents of*destgraph*are deleted by this operation.This operation is in effect equivalent to

*destgraph***deserialize**[*graphName***serialize**]

The operation assumes that the *destgraph* provides the method **deserialize** and that this method takes a graph serialization.

*graphName***append***key value*Appends a

*value*to one of the keyed values associated with the graph. Returns the new value given to the attribute*key*.*graphName***deserialize***serialization*This is the complement to

**serialize**. It replaces the graph data in*graphName*with the graph described by the*serialization*value. The old contents of*graphName*are deleted by this operation.*graphName***destroy**Destroys the graph, including its storage space and associated command.

*graphName***arc append***arc key value*Appends a

*value*to one of the keyed values associated with an*arc*. Returns the new value given to the attribute*key*.*graphName***arc attr***key**graphName***arc attr***key***-arcs***list**graphName***arc attr***key***-glob***globpattern**graphName***arc attr***key***-regexp***repattern*This method retrieves the value of the attribute named

*key*, for all arcs in the graph (matching the restriction specified via one of the possible options) and having the specified attribute.The result is a dictionary mapping from arc names to the value of attribute

*key*at that arc. Arcs not having the attribute*key*, or not passing a specified restriction, are not listed in the result.The possible restrictions are:

**-arcs**The value is a list of arcs. Only the arcs mentioned in this list are searched for the attribute.

**-glob**The value is a glob pattern. Only the arcs in the graph whose names match this pattern are searched for the attribute.

**-regexp**The value is a regular expression. Only the arcs in the graph whose names match this pattern are searched for the attribute.

*graphName***arc delete***arc*?*arc*...?Remove the specified arcs from the graph.

*graphName***arc exists***arc*Return true if the specified

*arc*exists in the graph.*graphName***arc flip***arc*Reverses the direction of the named

*arc*, i.e. the source and target nodes of the arc are exchanged with each other.*graphName***arc get***arc key*Returns the value associated with the key

*key*for the*arc*.*graphName***arc getall***arc*?*pattern*?Returns a dictionary (suitable for use with [

**array set**]) for the*arc*. If the*pattern*is specified only the attributes whose names match the pattern will be part of the returned dictionary. The pattern is a**glob**pattern.*graphName***arc getunweighted**Returns a list containing the names of all arcs in the graph which have no weight associated with them.

*graphName***arc getweight***arc*Returns the weight associated with the

*arc*. Throws an error if the arc has no weight associated with it.*graphName***arc keys***arc*?*pattern*?Returns a list of keys for the

*arc*. If the*pattern*is specified only the attributes whose names match the pattern will be part of the returned list. The pattern is a**glob**pattern.*graphName***arc keyexists***arc key*Return true if the specified

*key*exists for the*arc*.*graphName***arc insert***start end*?*child*?Insert an arc named

*child*into the graph beginning at the node*start*and ending at the node*end*. If the name of the new arc is not specified the system will generate a unique name of the form*arcx*.*graphName***arc lappend***arc key value*Appends a

*value*(as a list) to one of the keyed values associated with an*arc*. Returns the new value given to the attribute*key*.*graphName***arc rename***arc newname*Renames the arc

*arc*to*newname*. An error is thrown if either the arc does not exist, or a arc with name*newname*does exist. The result of the command is the new name of the arc.*graphName***arc set***arc key*?*value*?Set or get one of the keyed values associated with an arc. An arc may have any number of keyed values associated with it. If

*value*is not specified, this command returns the current value assigned to the key; if*value*is specified, this command assigns that value to the key, and returns that value.*graphName***arc setunweighted**?*weight*?Sets the weight of all arcs without a weight to

*weight*. Returns the empty string as its result. If not present*weight*defaults to**0**.*graphName***arc setweight***arc weight*Sets the weight of the

*arc*to*weight*. Returns*weight*.*graphName***arc unsetweight***arc*Removes the weight of the

*arc*, if present. Does nothing otherwise. Returns the empty string.*graphName***arc hasweight***arc*Determines if the

*arc*has a weight associated with it. The result is a boolean value,**True**if a weight is defined, and**False**otherwise.*graphName***arc source***arc*Return the node the given

*arc*begins at.*graphName***arc target***arc*Return the node the given

*arc*ends at.*graphName***arc nodes***arc*Return the nodes the given

*arc*begins and ends at, as a two-element list.*graphName***arc move-source***arc newsource*Changes the source node of the arc to

*newsource*. It can be said that the arc rotates around its target node.*graphName***arc move-target***arc newtarget*Changes the target node of the arc to

*newtarget*. It can be said that the arc rotates around its source node.*graphName***arc move***arc newsource newtarget*Changes both source and target nodes of the arc to

*newsource*, and*newtarget*resp.*graphName***arc unset***arc key*Remove a keyed value from the arc

*arc*. The method will do nothing if the*key*does not exist.*graphName***arc weights**Returns a dictionary whose keys are the names of all arcs which have a weight associated with them, and the values are these weights.

*graphName***arcs**?-key*key*? ?-value*value*? ?-filter*cmdprefix*? ?-in|-out|-adj|-inner|-embedding*node node...*?Returns a list of arcs in the graph. If no restriction is specified a list containing all arcs is returned. Restrictions can limit the list of returned arcs based on the nodes that are connected by the arc, on the keyed values associated with the arc, or both. A general filter command can be used as well. The restrictions that involve connected nodes take a variable number of nodes as argument, specified after the name of the restriction itself.

The restrictions imposed by either

**-in**,**-out**,**-adj**,**-inner**, or**-embedded**are applied first. Specifying more than one of them is illegal.After that the restrictions set via

**-key**(and**-value**) are applied. Specifying more than one**-key**(and**-value**) is illegal. Specifying**-value**alone, without**-key**is illegal as well.Any restriction set through

**-filter**is applied last. Specifying more than one**-filter**is illegal.Coming back to the restrictions based on a set of nodes, the command recognizes the following switches:

**-in**Return a list of all arcs whose target is one of the nodes in the set of nodes. I.e. it computes the union of all incoming arcs of the nodes in the set.

**-out**Return a list of all arcs whose source is one of the nodes in the set of nodes. I.e. it computes the union of all outgoing arcs of the nodes in the set.

**-adj**Return a list of all arcs adjacent to at least one of the nodes in the set. This is the union of the nodes returned by

**-in**and**-out**.**-inner**Return a list of all arcs which are adjacent to two of the nodes in the set. This is the set of arcs in the subgraph spawned by the specified nodes.

**-embedding**Return a list of all arcs adjacent to exactly one of the nodes in the set. This is the set of arcs connecting the subgraph spawned by the specified nodes to the rest of the graph.

**-key***key*Limit the list of arcs that are returned to those arcs that have an associated key

*key*.**-value***value*This restriction can only be used in combination with

**-key**. It limits the list of arcs that are returned to those arcs whose associated key*key*has the value*value*.**-filter***cmdrefix*Limit the list of arcs that are returned to those arcs that pass the test. The command in

*cmdprefix*is called with two arguments, the name of the graph object, and the name of the arc in question. It is executed in the context of the caller and has to return a boolean value. Arcs for which the command returns**false**are removed from the result list before it is returned to the caller.

*graphName***lappend***key value*Appends a

*value*(as a list) to one of the keyed values associated with the graph. Returns the new value given to the attribute*key*.*graphName***node append***node key value*Appends a

*value*to one of the keyed values associated with an*node*. Returns the new value given to the attribute*key*.*graphName***node attr***key**graphName***node attr***key***-nodes***list**graphName***node attr***key***-glob***globpattern**graphName***node attr***key***-regexp***repattern*This method retrieves the value of the attribute named

*key*, for all nodes in the graph (matching the restriction specified via one of the possible options) and having the specified attribute.The result is a dictionary mapping from node names to the value of attribute

*key*at that node. Nodes not having the attribute*key*, or not passing a specified restriction, are not listed in the result.The possible restrictions are:

**-nodes**The value is a list of nodes. Only the nodes mentioned in this list are searched for the attribute.

**-glob**The value is a glob pattern. Only the nodes in the graph whose names match this pattern are searched for the attribute.

**-regexp**The value is a regular expression. Only the nodes in the graph whose names match this pattern are searched for the attribute.

*graphName***node degree**?-in|-out?*node*Return the number of arcs adjacent to the specified

*node*. If one of the restrictions**-in**or**-out**is given only the incoming resp. outgoing arcs are counted.*graphName***node delete***node*?*node*...?Remove the specified nodes from the graph. All of the nodes' arcs will be removed as well to prevent unconnected arcs.

*graphName***node exists***node*Return true if the specified

*node*exists in the graph.*graphName***node get***node key*Return the value associated with the key

*key*for the*node*.*graphName***node getall***node*?*pattern*?Returns a dictionary (suitable for use with [

**array set**]) for the*node*. If the*pattern*is specified only the attributes whose names match the pattern will be part of the returned dictionary. The pattern is a**glob**pattern.*graphName***node keys***node*?*pattern*?Returns a list of keys for the

*node*. If the*pattern*is specified only the attributes whose names match the pattern will be part of the returned list. The pattern is a**glob**pattern.*graphName***node keyexists***node key*Return true if the specified

*key*exists for the*node*.*graphName***node insert**?*node*...?Insert one or more nodes into the graph. The new nodes have no arcs connected to them. If no node is specified one node will be inserted, and the system will generate a unique name of the form

*nodex*for it.*graphName***node lappend***node key value*Appends a

*value*(as a list) to one of the keyed values associated with an*node*. Returns the new value given to the attribute*key*.*graphName***node opposite***node arc*Return the node at the other end of the specified

*arc*, which has to be adjacent to the given*node*.*graphName***node rename***node newname*Renames the node

*node*to*newname*. An error is thrown if either the node does not exist, or a node with name*newname*does exist. The result of the command is the new name of the node.*graphName***node set***node key*?*value*?Set or get one of the keyed values associated with a node. A node may have any number of keyed values associated with it. If

*value*is not specified, this command returns the current value assigned to the key; if*value*is specified, this command assigns that value to the key.*graphName***node unset***node key*Remove a keyed value from the node

*node*. The method will do nothing if the*key*does not exist.*graphName***nodes**?-key*key*? ?-value*value*? ?-filter*cmdprefix*? ?-in|-out|-adj|-inner|-embedding*node node*...?Return a list of nodes in the graph. Restrictions can limit the list of returned nodes based on neighboring nodes, or based on the keyed values associated with the node. The restrictions that involve neighboring nodes have a list of nodes as argument, specified after the name of the restriction itself.

The possible restrictions are the same as for method

**arcs**. The exact meanings change slightly, as they operate on nodes instead of arcs. The command recognizes:**-in**Return a list of all nodes with at least one outgoing arc ending in a node found in the specified set of nodes. Alternatively specified as the set of source nodes for the

**-in**arcs of the node set. The*incoming neighbours*.**-out**Return a list of all nodes with at least one incoming arc starting in a node found in the specified set of nodes. Alternatively specified as the set of target nodes for the

**-out**arcs of the node set. The*outgoing neighbours*.**-adj**This is the union of the nodes returned by

**-in**and**-out**. The*neighbours*.**-inner**The set of neighbours (see

**-adj**above) which are also in the set of nodes. I.e. the intersection between the set of nodes and the neighbours per**-adj**.**-embedding**The set of neighbours (see

**-adj**above) which are not in the set of nodes. I.e. the difference between the neighbours as per**-adj**, and the set of nodes.**-key***key*Limit the list of nodes that are returned to those nodes that have an associated key

*key*.**-value***value*This restriction can only be used in combination with

**-key**. It limits the list of nodes that are returned to those nodes whose associated key*key*has the value*value*.**-filter***cmdrefix*Limit the list of nodes that are returned to those nodes that pass the test. The command in

*cmdprefix*is called with two arguments, the name of the graph object, and the name of the node in question. It is executed in the context of the caller and has to return a boolean value. Nodes for which the command returns**false**are removed from the result list before it is returned to the caller.

*graphName***get***key*Return the value associated with the key

*key*for the graph.*graphName***getall**?*pattern*?Returns a dictionary (suitable for use with [

**array set**]) for the whole graph. If the*pattern*is specified only the attributes whose names match the pattern will be part of the returned dictionary. The pattern is a**glob**pattern.*graphName***keys**?*pattern*?Returns a list of keys for the whole graph. If the

*pattern*is specified only the attributes whose names match the pattern will be part of the returned list. The pattern is a**glob**pattern.*graphName***keyexists***key*Return true if the specified

*key*exists for the whole graph.*graphName***serialize**?*node*...?This method serializes the sub-graph spanned up by the

*node*s. In other words it returns a tcl value completely describing that graph. If no nodes are specified the whole graph will be serialized. This allows, for example, the transfer of graph objects (or parts thereof) over arbitrary channels, persistence, etc. This method is also the basis for both the copy constructor and the assignment operator.The result of this method has to be semantically identical over all implementations of the graph interface. This is what will enable us to copy graph data between different implementations of the same interface.

The result is a list containing a multiple of three items, plus one! In other words, '[llength $serial] % 3 == 1'. Valid values include 1, 4, 7, ...

The last element of the list is a dictionary containing the attributes associated with the whole graph. Regarding the other elements; each triple consists of

- [1]
The name of the node to be described,

- [2]
A dictionary containing the attributes associated with the node,

- [3]
And a list describing all the arcs starting at that node.

The elements of the arc list are lists containing three or four elements each, i.e.

- [1]
The name of the arc described by the element,

- [2]
A reference to the destination node of the arc. This reference is an integer number given the index of that node in the main serialization list. As that it is greater than or equal to zero, less than the length of the serialization, and a multiple of three.

*Note:*For internal consistency no arc name may be used twice, whether in the same node, or at some other node. This is a global consistency requirement for the serialization.- [3]
And a dictionary containing the attributes associated with the arc.

- [4]
The weight associated with the arc. This value is optional. Its non-presence means that the arc in question has no weight associated with it.

*Note:*This information is new, compared to the serialization of**graph**2.3 and earlier. By making it an optional element the new format is maximally compatible with the old. This means that any graph not using weights will generate a serialization which is still understood by the older graph package. A serialization will not be understood any longer by the older packages if, and only if the graph it was generated from actually has arcs with weights.

For all attribute dictionaries they keys are the names of the attributes, and the values are the values for each name.

*Note:* The order of the nodes in the serialization has no relevance, nor has the order of the arcs per node.

# A possible serialization for the graph structure # # d -----> %2 # / ^ \\ # / / \\ # / b \\ # / / \\ # %1 <- a - %0 e # ^ \\ / # \\ c / # \\ \\ / # \\ v v # f ------ %3 # is # # %3 {} {{f 6 {}}} %0 {} {{a 6 {}} {b 9 {}} {c 0 {}}} %1 {} {{d 9 {}}} %2 {} {{e 0 {}}} {} # # This assumes that the graph has neither attribute data nor weighted arcs.

*graphName***set***key*?*value*?Set or get one of the keyed values associated with a graph. A graph may have any number of keyed values associated with it. If

*value*is not specified, this command returns the current value assigned to the key; if*value*is specified, this command assigns that value to the key.*graphName***swap***node1 node2*Swap the position of

*node1*and*node2*in the graph.*graphName***unset***key*Remove a keyed value from the graph. The method will do nothing if the

*key*does not exist.*graphName***walk***node*?-order*order*? ?-type*type*? ?-dir*direction*? -command*cmd*Perform a breadth-first or depth-first walk of the graph starting at the node

*node*going in either the direction of outgoing or opposite to the incoming arcs.The type of walk, breadth-first or depth-first, is determined by the value of

*type*;**bfs**indicates breadth-first,**dfs**indicates depth-first. Depth-first is the default.The order of the walk, pre-order, post-order or both-order is determined by the value of

*order*;**pre**indicates pre-order,**post**indicates post-order,**both**indicates both-order. Pre-order is the default. Pre-order walking means that a node is visited before any of its neighbors (as defined by the*direction*, see below). Post-order walking means that a parent is visited after any of its neighbors. Both-order walking means that a node is visited before*and*after any of its neighbors. The combination of a breadth-first walk with post- or both-order is illegal.The direction of the walk is determined by the value of

*dir*;**backward**indicates the direction opposite to the incoming arcs,**forward**indicates the direction of the outgoing arcs.As the walk progresses, the command

*cmd*will be evaluated at each node, with the mode of the call (**enter**or**leave**) and values*graphName*and the name of the current node appended. For a pre-order walk, all nodes are**enter**ed, for a post-order all nodes are left. In a both-order walk the first visit of a node**enter**s it, the second visit**leave**s it.

## Changes for 2.0

The following noteworthy changes have occurred:

- [1]
The API for accessing attributes and their values has been simplified.

All functionality regarding the default attribute "data" has been removed. This default attribute does not exist anymore. All accesses to attributes have to specify the name of the attribute in question. This backward

*incompatible*change allowed us to simplify the signature of all methods handling attributes.Especially the flag

**-key**is not required anymore, even more, its use is now forbidden. Please read the documentation for the arc and node methods**set**,**get**,**getall**,**unset**,**append**,**lappend**,**keyexists**and**keys**for a description of the new API's.- [2]
The methods

**keys**and**getall**now take an optional pattern argument and will return only attribute data for keys matching this pattern.- [3]
Arcs and nodes can now be renamed. See the documentation for the methods

**arc rename**and**node rename**.- [4]
The structure has been extended with API's for the serialization and deserialization of graph objects, and a number of operations based on them (graph assignment, copy construction).

Please read the documentation for the methods

**serialize**,**deserialize**,**=**, and**-->**, and the documentation on the construction of graph objects.Beyond the copying of whole graph objects these new API's also enable the transfer of graph objects over arbitrary channels and for easy persistence.

- [5]
A new method,

**attr**, was added to both**arc**and**node**allowing the query and retrieval of attribute data without regard to arc and node relationships.- [6]
Both methods

**arcs**and**nodes**have been extended with the ability to select arcs and nodes based on an arbitrary filtering criterium.

## Bugs, Ideas, Feedback

This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category *struct :: graph* of the *Tcllib Trackers* [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for enhancements you may have for either package and/or documentation.

When proposing code changes, please provide *unified diffs*, i.e the output of **diff -u**.

Note further that *attachments* are strongly preferred over inlined patches. Attachments can be made by going to the **Edit** form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar.

## Keywords

adjacent, arc, cgraph, degree, edge, graph, loop, neighbour, node, serialization, subgraph, vertex

## Category

Data structures

## Copyright

Copyright (c) 2002-2009 Andreas Kupries <andreas_kupries@users.sourceforge.net>