modules/struct/graphops.man

[comment {-*- tcl -*-}]
[manpage_begin struct::graph::op n 0.11.3]
[copyright {2008 Alejandro Paz <vidriloco@gmail.com>}]
[copyright {2008 (docs) Andreas Kupries <andreas_kupries@users.sourceforge.net>}]
[copyright {2009 Michal Antoniewski <antoniewski.m@gmail.com>}]
[moddesc   {Tcl Data Structures}]
[titledesc {Operation for (un)directed graph objects}]
[category  {Data structures}]
[require Tcl 8.4]
[require struct::graph::op [opt 0.11.3]]
[comment {[require struct::graph [opt 2.3]]   }]
[comment {[require struct::list  [opt 1.5]]   }]
[comment {[require struct::set   [opt 2.2.3]] }]
[description]
[para]

The package described by this document, [package struct::graph::op],
is a companion to the package [package struct::graph]. It provides a
series of common operations and algorithms applicable to (un)directed
graphs.

[para]

Despite being a companion the package is not directly dependent on
[package struct::graph], only on the API defined by that
package. I.e. the operations of this package can be applied to any and
all graph objects which provide the same API as the objects created
through [package struct::graph].

[section {Operations}]

[list_begin definitions]

[call [cmd struct::graph:op::toAdjacencyMatrix] [arg g]]

This command takes the graph [arg g] and returns a nested list
containing the adjacency matrix of [arg g].

[para]

The elements of the outer list are the rows of the matrix, the inner
elements are the column values in each row. The matrix has "[var n]+1"
rows and columns, with the first row and column (index 0) containing
the name of the node the row/column is for. All other elements are
boolean values, [const True] if there is an arc between the 2 nodes
of the respective row and column, and [const False] otherwise.

[para]

Note that the matrix is symmetric. It does not represent the
directionality of arcs, only their presence between nodes. It is also
unable to represent parallel arcs in [arg g].


[call [cmd struct::graph:op::toAdjacencyList] [arg G] [opt [arg options]...]]

Procedure creates for input graph [arg G], it's representation as [term "Adjacency List"].
It handles both directed and undirected graphs (default is undirected).
It returns dictionary that for each node (key) returns list of nodes adjacent
to it. When considering weighted version, for each adjacent node there is also
weight of the edge included.

[para]
[list_begin definitions]
[def Arguments:]

[list_begin arguments]
[arg_def {Graph object} G input]
A graph to convert into an [term "Adjacency List"].

[list_end][comment {-- arguments --}]

[def Options:]
[list_begin options]
[opt_def -directed]

By default [arg G] is operated as if it were an [term {Undirected graph}].
Using this option tells the command to handle [arg G] as the directed graph it is.

[opt_def -weights]

By default any weight information the graph [arg G] may have is ignored.
Using this option tells the command to put weight information into the result.
In that case it is expected that all arcs have a proper weight, and an error
is thrown if that is not the case.

[list_end][comment {-- options --}]
[list_end][comment {-- definitions --}]


[call [cmd struct::graph:op::kruskal] [arg g]]

This command takes the graph [arg g] and returns a list containing the
names of the arcs in [arg g] which span up a minimum weight spanning tree
(MST), or, in the case of an un-connected graph, a minimum weight spanning
forest (except for the 1-vertex components). Kruskal's algorithm is used
to compute the tree or forest.

This algorithm has a time complexity of [term {O(E*log E)}] or [term {O(E* log V)}],
where [term V] is the number of vertices and [term E] is the number of edges
in graph [arg g].

[para]

The command will throw an error if one or more arcs in [arg g] have no
weight associated with them.

[para]

A note regarding the result, the command refrains from explicitly
listing the nodes of the MST as this information is implicitly
provided in the arcs already.


[call [cmd struct::graph:op::prim] [arg g]]

This command takes the graph [arg g] and returns a list containing the
names of the arcs in [arg g] which span up a minimum weight spanning tree
(MST), or, in the case of an un-connected graph, a minimum weight spanning
forest (except for the 1-vertex components). Prim's algorithm is used to
compute the tree or forest.

This algorithm has a time complexity between [term {O(E+V*log V)}] and [term {O(V*V)}],
depending on the implementation (Fibonacci heap + Adjacency list versus
Adjacency Matrix).  As usual [term V] is the number of vertices and
[term E] the number of edges in graph [arg g].

[para]

The command will throw an error if one or more arcs in [arg g] have no
weight associated with them.

[para]

A note regarding the result, the command refrains from explicitly
listing the nodes of the MST as this information is implicitly
provided in the arcs already.


[call [cmd struct::graph:op::isBipartite?] [arg g] [opt [arg bipartvar]]]

This command takes the graph [arg g] and returns a boolean value
indicating whether it is bipartite ([const true]) or not
([const false]). If the variable [arg bipartvar] is specified the two
partitions of the graph are there as a list, if, and only if the graph
is bipartit. If it is not the variable, if specified, is not touched.


[call [cmd struct::graph:op::tarjan] [arg g]]

This command computes the set of [emph {strongly connected}]
components (SCCs) of the graph [arg g]. The result of the command is a
list of sets, each of which contains the nodes for one of the SCCs of
[arg g]. The union of all SCCs covers the whole graph, and no two SCCs
intersect with each other.

[para]

The graph [arg g] is [term acyclic] if all SCCs in the result contain
only a single node. The graph [arg g] is [term {strongly connected}]
if the result contains only a single SCC containing all nodes of
[arg g].


[call [cmd struct::graph:op::connectedComponents] [arg g]]

This command computes the set of [emph connected] components (CCs) of
the graph [arg g]. The result of the command is a list of sets, each
of which contains the nodes for one of the CCs of [arg g]. The union
of all CCs covers the whole graph, and no two CCs intersect with each
other.

[para]

The graph [arg g] is [term connected] if the result contains only a
single SCC containing all nodes of [arg g].


[call [cmd struct::graph:op::connectedComponentOf] [arg g] [arg n]]

This command computes the [emph connected] component (CC) of the graph
[arg g] containing the node [arg n]. The result of the command is a
sets which contains the nodes for the CC of [arg n] in [arg g].

[para]

The command will throw an error if [arg n] is not a node of the graph
[arg g].


[call [cmd struct::graph:op::isConnected?] [arg g]]

This is a convenience command determining whether the graph [arg g] is
[term connected] or not.  The result is a boolean value, [const true]
if the graph is connected, and [const false] otherwise.


[call [cmd struct::graph:op::isCutVertex?] [arg g] [arg n]]

This command determines whether the node [arg n] in the graph [arg g]
is a [term {cut vertex}] (aka [term {articulation point}]). The result
is a boolean value, [const true] if the node is a cut vertex, and
[const false] otherwise.

[para]

The command will throw an error if [arg n] is not a node of the graph
[arg g].


[call [cmd struct::graph:op::isBridge?] [arg g] [arg a]]

This command determines whether the arc [arg a] in the graph [arg g]
is a [term bridge] (aka [term {cut edge}], or [term isthmus]). The
result is a boolean value, [const true] if the arc is a bridge, and
[const false] otherwise.

[para]

The command will throw an error if [arg a] is not an arc of the graph
[arg g].


[call [cmd struct::graph:op::isEulerian?] [arg g] [opt [arg tourvar]]]

This command determines whether the graph [arg g] is [term eulerian]
or not.  The result is a boolean value, [const true] if the graph is
eulerian, and [const false] otherwise.

[para]

If the graph is eulerian and [arg tourvar] is specified then an euler
tour is computed as well and stored in the named variable. The tour is
represented by the list of arcs traversed, in the order of traversal.


[call [cmd struct::graph:op::isSemiEulerian?] [arg g] [opt [arg pathvar]]]

This command determines whether the graph [arg g] is
[term semi-eulerian] or not.  The result is a boolean value, [const true]
if the graph is semi-eulerian, and [const false] otherwise.

[para]

If the graph is semi-eulerian and [arg pathvar] is specified then an
euler path is computed as well and stored in the named variable. The
path is represented by the list of arcs traversed, in the order of
traversal.


[call [cmd struct::graph:op::dijkstra] [arg g] [arg start] [opt [arg options]...]]

This command determines distances in the weighted [arg g] from the
node [arg start] to all other nodes in the graph. The options specify
how to traverse graphs, and the format of the result.

[para]

Two options are recognized

[list_begin options]
[opt_def -arcmode mode]

The accepted mode values are [const directed] and [const undirected].
For directed traversal all arcs are traversed from source to
target. For undirected traversal all arcs are traversed in the
opposite direction as well. Undirected traversal is the default.

[opt_def -outputformat format]

The accepted format values are [const distances] and [const tree]. In
both cases the result is a dictionary keyed by the names of all nodes
in the graph. For [const distances] the value is the distance of the
node to [arg start], whereas for [const tree] the value is the path
from the node to [arg start], excluding the node itself, but including
[arg start]. Tree format is the default.

[list_end]


[call [cmd struct::graph:op::distance] [arg g] [arg origin] [arg destination] [opt [arg options]...]]

This command determines the (un)directed distance between the two
nodes [arg origin] and [arg destination] in the graph [arg g]. It
accepts the option [option -arcmode] of [cmd struct::graph:op::dijkstra].


[call [cmd struct::graph:op::eccentricity] [arg g] [arg n] [opt [arg options]...]]

This command determines the (un)directed [term eccentricity] of the
node [arg n] in the graph [arg g]. It accepts the option
[option -arcmode] of [cmd struct::graph:op::dijkstra].

[para]

The (un)directed [term eccentricity] of a node is the maximal
(un)directed distance between the node and any other node in the
graph.


[call [cmd struct::graph:op::radius] [arg g] [opt [arg options]...]]

This command determines the (un)directed [term radius] of the graph
[arg g]. It accepts the option [option -arcmode] of [cmd struct::graph:op::dijkstra].

[para]

The (un)directed [term radius] of a graph is the minimal (un)directed
[term eccentricity] of all nodes in the graph.


[call [cmd struct::graph:op::diameter] [arg g] [opt [arg options]...]]

This command determines the (un)directed [term diameter] of the graph
[arg g]. It accepts the option [option -arcmode] of [cmd struct::graph:op::dijkstra].

[para]

The (un)directed [term diameter] of a graph is the maximal (un)directed
[term eccentricity] of all nodes in the graph.


[call [cmd struct::graph::op::BellmanFord] [arg G] [arg startnode]]

Searching for [sectref {Shortest Path Problem} "shortests paths"] between chosen node and all other nodes in graph [arg G]. Based
on relaxation method. In comparison to [cmd struct::graph::op::dijkstra] it doesn't need assumption that all weights
on edges in input graph [arg G] have to be positive.

[para]

That generality sets the complexity of algorithm to - [term O(V*E)], where [term V] is the number of vertices
and [term E] is number of edges in graph [arg G].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph object} G input]
Directed, connected and edge weighted graph [arg G], without any negative cycles ( presence of cycles with the negative sum
of weight means that there is no shortest path, since the total weight becomes lower each time the cycle is
traversed ). Negative weights on edges are allowed.

[arg_def {Node} startnode input]
The node for which we find all shortest paths to each other node in graph [arg G].
[list_end][comment {-- arguments --}]

[def Result:]
Dictionary containing for each node (key) distances to each other node in graph [arg G].
[list_end][comment {-- definitions --}]

[para]

[emph Note:] If algorithm finds a negative cycle, it will return error message.


[call [cmd struct::graph::op::Johnsons] [arg G] [opt [arg options]...]]

Searching for [sectref {Shortest Path Problem} "shortest paths"] between all pairs of vertices in graph. For sparse graphs
asymptotically quicker than [cmd struct::graph::op::FloydWarshall] algorithm. Johnson's algorithm
uses [cmd struct::graph::op::BellmanFord] and [cmd struct::graph::op::dijkstra] as subprocedures.

[para]

Time complexity: [term "O(n**2*log(n) +n*m)"], where [term n] is the number of nodes and [term m] is
the number of edges in graph [arg G].

[para]

[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph object} G input]

Directed graph [arg G], weighted on edges and not containing
any cycles with negative sum of weights ( the presence of such cycles means
there is no shortest path, since the total weight becomes lower each time the
cycle is traversed ). Negative weights on edges are allowed.

[list_end][comment {-- arguments --}]

[def Options:]
[list_begin options]
[opt_def -filter]

Returns only existing distances, cuts all [term Inf] values for
non-existing connections between pairs of nodes.

[list_end][comment {-- options --}]

[def Result:]

Dictionary containing distances between all pairs of vertices.

[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::FloydWarshall] [arg G]]

Searching for [sectref {Shortest Path Problem} "shortest paths"] between all pairs of edges in weighted graphs.[para]
Time complexity: [term O(V^3)] - where [term V] is number of vertices.[para]
Memory complexity: [term O(V^2)].

[para]

[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph object} G input]

Directed and weighted graph [arg G].

[list_end][comment {-- arguments --}]

[def Result:]
Dictionary containing shortest distances to each node from each node.
[list_end][comment {-- definitions --}]

[emph Note:] Algorithm finds solutions dynamically. It compares all possible paths through the graph
between each pair of vertices. Graph shouldn't possess any cycle with negative
sum of weights (the presence of such cycles means there is no shortest path,
since the total weight becomes lower each time the cycle is traversed).
[para]
On the other hand algorithm can be used to find those cycles - if any shortest distance
found by algorithm for any nodes [term v] and [term u] (when [term v] is the same node as [term u]) is negative,
that node surely belong to at least one negative cycle.


[call [cmd struct::graph::op::MetricTravellingSalesman] [arg G]]

Algorithm for solving a metric variation of [sectref {Travelling Salesman Problem} "Travelling salesman problem"].
[term "TSP problem"] is [term NP-Complete], so there is no efficient algorithm to solve it. Greedy methods
are getting extremely slow, with the increase in the set of nodes.

[para]

[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph object} G input]
Undirected, weighted graph [arg G].
[list_end][comment {-- arguments --}]

[def Result:]
Approximated solution of minimum [term "Hamilton Cycle"] - closed path visiting all nodes,
each exactly one time.

[list_end][comment {-- definitions --}]

[emph Note:] [sectref {Approximation algorithm} "It's 2-approximation algorithm."]


[call [cmd struct::graph::op::Christofides] [arg G]]

Another algorithm for solving [sectref {Travelling Salesman Problem} "metric [term "TSP problem"]"].
Christofides implementation uses [term "Max Matching"] for reaching better approximation factor.

[para]

[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]

Undirected, weighted graph [arg G].

[list_end][comment {-- arguments --}]

[def Result:]
Approximated solution of minimum [term "Hamilton Cycle"] - closed path visiting all nodes,
each exactly one time.

[list_end][comment {-- definitions --}]

[para]

[emph Note:] [sectref {Approximation algorithm} "It's is a 3/2 approximation algorithm. "]


[call [cmd struct::graph::op::GreedyMaxMatching] [arg G]]

[term "Greedy Max Matching"] procedure, which finds [sectref {Matching Problem} "maximal matching"] (not maximum)
for given graph [arg G]. It adds edges to solution, beginning from edges with the
lowest cost.

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]

Undirected graph [arg G].

[list_end][comment {-- arguments --}]

[def Result:]
Set of edges - the max matching for graph [arg G].

[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::MaxCut] [arg G] [arg U] [arg V]]

Algorithm solving a [sectref {Cut Problems} "Maximum Cut Problem"].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]

The graph to cut.

[arg_def {List} U output]

Variable storing first set of nodes (cut) given by solution.

[arg_def {List} V output]

Variable storing second set of nodes (cut) given by solution.

[list_end][comment {-- arguments --}]

[def Result:]
Algorithm returns number of edges between found two sets of nodes.

[list_end][comment {-- definitions --}]

[emph Note:] [term MaxCut] is a [sectref {Approximation algorithm} "2-approximation algorithm."]


[call [cmd struct::graph::op::UnweightedKCenter] [arg G] [arg k]]

Approximation algorithm that solves a [sectref {K-Center Problem} "k-center problem"].


[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Undirected complete graph [arg G], which satisfies triangle inequality.[para]
[arg_def {Integer} k input]
Positive integer that sets the number of nodes that will be included in [term "k-center"].

[list_end][comment {-- arguments --}]

[def Result:]
Set of nodes - [arg k] center for graph [arg G].

[list_end][comment {-- definitions --}]

[emph Note:] [term UnweightedKCenter] is a [sectref {Approximation algorithm} "2-approximation algorithm."]


[call [cmd struct::graph::op::WeightedKCenter] [arg G] [arg nodeWeights] [arg W]]

Approximation algorithm that solves a weighted version of [sectref {K-Center Problem} "k-center problem"].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Undirected complete graph [arg G], which satisfies triangle inequality.
[arg_def {Integer} W input]
Positive integer that sets the maximum possible weight of [term "k-center"] found by algorithm.
[arg_def {List} nodeWeights input]
List of nodes and its weights in graph [arg G].

[list_end][comment {-- arguments --}]

[def Result:]
Set of nodes, which is solution found by algorithm.
[list_end][comment {-- definitions --}]

[emph Note:][term WeightedKCenter] is a [sectref {Approximation algorithm} "3-approximation algorithm."]


[call [cmd struct::graph::op::GreedyMaxIndependentSet] [arg G]]

A [term "maximal independent set"] is an [term "independent set"] such that adding any other node
to the set forces the set to contain an edge.

[para]

Algorithm for input graph [arg G] returns set of nodes (list), which are contained in Max Independent
Set found by algorithm.


[call [cmd struct::graph::op::GreedyWeightedMaxIndependentSet] [arg G] [arg nodeWeights]]

Weighted variation of [term "Maximal Independent Set"]. It takes as an input argument
not only graph [arg G] but also set of weights for all vertices in graph [arg G].

[para]
[emph Note:]
Read also [term "Maximal Independent Set"] description for more info.


[call [cmd struct::graph::op::VerticesCover] [arg G]]

[term "Vertices cover"] is a set of vertices such that each edge of the graph is incident to
at least one vertex of the set. This 2-approximation algorithm searches for minimum
[term "vertices cover"], which is a classical optimization problem in computer science and
is a typical example of an [term "NP-hard"] optimization problem that has an approximation
algorithm.

For input graph [arg G] algorithm returns the set of edges (list), which is Vertex Cover found by algorithm.


[call [cmd struct::graph::op::EdmondsKarp] [arg G] [arg s] [arg t]]

Improved Ford-Fulkerson's algorithm, computing the [sectref {Flow Problems} "maximum flow"] in given flow network [arg G].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Weighted and directed graph. Each edge should have set integer attribute considered as
maximum throughputs that can be carried by that link (edge).
[arg_def {Node} s input]
The node that is a source for graph [arg G].
[arg_def {Node} t input]
The node that is a sink for graph [arg G].
[list_end][comment {-- arguments --}]

[def Result:]
Procedure returns the dictionary containing throughputs for all edges. For
each key ( the edge between nodes [term u] and [term v] in the form of [term "list u v"] ) there is
a value that is a throughput for that key. Edges where throughput values
are equal to 0 are not returned ( it is like there was no link in the flow network
between nodes connected by such edge).

[list_end][comment {-- definitions --}]

[para]

The general idea of algorithm is finding the shortest augumenting paths in graph [arg G], as long
as they exist, and for each path updating the edge's weights along that path,
with maximum possible throughput. The final (maximum) flow is found
when there is no other augumenting path from source to sink.

[para]

[emph Note:] Algorithm complexity : [term O(V*E)], where [term V] is the number of nodes and [term E] is the number
of edges in graph [term G].


[call [cmd struct::graph::op::BusackerGowen] [arg G] [arg desiredFlow] [arg s] [arg t]]

Algorithm finds solution for a [sectref {Flow Problems} "minimum cost flow problem"]. So, the goal is to find a flow,
whose max value can be [arg desiredFlow], from source node [arg s] to sink node [arg t] in given flow network [arg G].
That network except throughputs at edges has also defined a non-negative cost on each edge - cost of using that edge when
directing flow with that edge ( it can illustrate e.g. fuel usage, time or any other measure dependent on usages ).

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Flow network (directed graph), each edge in graph should have two integer attributes: [term cost] and [term throughput].
[arg_def {Integer} desiredFlow input]
Max value of the flow for that network.
[arg_def {Node} s input]
The source node for graph [arg G].
[arg_def {Node} t input]
The sink node for graph [arg G].

[list_end][comment {-- arguments --}]

[def Result:]
Dictionary containing values of used throughputs for each edge ( key ).
found by algorithm.

[list_end][comment {-- definitions --}]

[emph Note:] Algorithm complexity : [term O(V**2*desiredFlow)], where [term V] is the number of nodes in graph [arg G].


[call [cmd struct::graph::op::ShortestsPathsByBFS] [arg G] [arg s] [arg outputFormat]]

Shortest pathfinding algorithm using BFS method. In comparison to [cmd struct::graph::op::dijkstra] it can
work with negative weights on edges. Of course negative cycles are not allowed. Algorithm is better than dijkstra
for sparse graphs, but also there exist some pathological cases (those cases generally don't appear in practise) that
make time complexity increase exponentially with the growth of the number of nodes.

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Input graph.
[arg_def {Node} s input]
Source node for which all distances to each other node in graph [arg G] are computed.
[list_end][comment {-- arguments --}]

[def "Options and result:"]
[list_begin options]
[opt_def distances]

When selected [arg outputFormat] is [const distances] - procedure returns dictionary containing
distances between source node [arg s] and each other node in graph [arg G].

[opt_def paths]

When selected [arg outputFormat] is [const paths] - procedure returns dictionary containing
for each node [term v], a list of nodes, which is a path between source node [arg s] and node [term v].

[list_end][comment {-- options --}]


[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::BFS] [arg G] [arg s] [opt [arg outputFormat]...]]

Breadth-First Search - algorithm creates the BFS Tree.
Memory and time complexity: [term "O(V + E)"], where [term V] is the number of nodes and [term E]
is number of edges.

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Input graph.
[arg_def {Node} s input]
Source node for BFS procedure.

[list_end][comment {-- arguments --}]

[def "Options and result:"]
[list_begin options]
[opt_def graph]

When selected [option outputFormat] is [option graph] - procedure returns a graph structure ([cmd struct::graph]),
which is equivalent to BFS tree found by algorithm.

[opt_def tree]

When selected [option outputFormat] is [option tree] - procedure returns a tree structure ([cmd struct::tree]),
which is equivalent to BFS tree found by algorithm.

[list_end][comment {-- options --}]

[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::MinimumDiameterSpanningTree] [arg G]]

The goal is to find for input graph [arg G], the [term "spanning tree"] that
has the minimum [term "diameter"] value.

[para]

General idea of algorithm is to run [term BFS] over all vertices in graph
[arg G]. If the diameter [term d] of the tree is odd, then we are sure that tree
given by [term BFS] is minimum (considering diameter value). When, diameter [term d]
is even, then optimal tree can have minimum [term diameter] equal to [term d] or
[term d-1].

[para]

In that case, what algorithm does is rebuilding the tree given by [term BFS], by
adding a vertice between root node and root's child node (nodes), such that
subtree created with child node as root node is the greatest one (has the
greatests height). In the next step for such rebuilded tree, we run again [term BFS]
with new node as root node. If the height of the tree didn't changed, we have found
a better solution.

[para]

For input graph [arg G] algorithm returns the graph structure ([cmd struct::graph]) that is
a spanning tree with minimum diameter found by algorithm.


[call [cmd struct::graph::op::MinimumDegreeSpanningTree] [arg G]]

Algorithm finds for input graph [arg G], a spanning tree [term T] with the minimum possible
degree. That problem is [term NP-hard], so algorithm is an approximation algorithm.

[para]

Let [term V] be the set of nodes for graph [arg G] and let [term W] be any subset of [term V]. Lets
assume also that [term OPT] is optimal solution and [term ALG] is solution found by algorithm for input
graph [arg G]. [para]
It can be proven that solution found with the algorithm must fulfil inequality: [para]  [term "((|W| + k - 1) / |W|) <= ALG <= 2*OPT + log2(n) + 1"].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Undirected simple graph.

[list_end][comment {-- arguments --}]

[def Result:]
Algorithm returns graph structure, which is equivalent to spanning tree [term T] found by algorithm.

[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::MaximumFlowByDinic] [arg G] [arg s] [arg t] [arg blockingFlowAlg]]

Algorithm finds [sectref {Flow Problems} "maximum flow"] for the flow network represented by graph [arg G]. It is based on
the blocking-flow finding methods, which give us different complexities what makes a better fit for
different graphs.

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Directed graph [arg G] representing the flow network. Each edge should have attribute
[term throughput] set with integer value.

[arg_def {Node} s input]
The source node for the flow network [arg G].

[arg_def {Node} t input]
The sink node for the flow network [arg G].

[list_end][comment {-- arguments --}]

[def Options:]
[list_begin options]
[opt_def dinic]
Procedure will find maximum flow for flow network [arg G] using Dinic's algorithm ([cmd struct::graph::op::BlockingFlowByDinic])
for blocking flow computation.
[opt_def mkm]
Procedure will find maximum flow for flow network [arg G] using Malhotra, Kumar and Maheshwari's algorithm ([cmd struct::graph::op::BlockingFlowByMKM])
for blocking flow computation.
[list_end][comment {-- options --}]

[def Result:]
Algorithm returns dictionary containing it's flow value for each edge (key) in network [arg G].

[list_end][comment {-- definitions --}]

[para]

[emph Note:] [cmd struct::graph::op::BlockingFlowByDinic] gives [term O(m*n^2)] complexity and
[cmd struct::graph::op::BlockingFlowByMKM] gives [term O(n^3)] complexity, where [term n] is the number of nodes
and [term m] is the number of edges in flow network [arg G].


[call [cmd struct::graph::op::BlockingFlowByDinic] [arg G] [arg s] [arg t]]

Algorithm for given network [arg G] with source [arg s] and sink [arg t], finds a [sectref {Flow Problems} "blocking
flow"], which can be used to obtain a [term "maximum flow"] for that network [arg G].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Directed graph [arg G] representing the flow network. Each edge should have attribute
[term throughput] set with integer value.
[arg_def {Node} s input]
The source node for the flow network [arg G].
[arg_def {Node} t input]
The sink node for the flow network [arg G].
[list_end][comment {-- arguments --}]

[def Result:]
Algorithm returns dictionary containing it's blocking flow value for each edge (key) in network [arg G].

[list_end][comment {-- definitions --}]

[emph Note:] Algorithm's complexity is [term O(n*m)], where [term n] is the number of nodes
and [term m] is the number of edges in flow network [arg G].


[call [cmd struct::graph::op::BlockingFlowByMKM] [arg G] [arg s] [arg t]]

Algorithm for given network [arg G] with source [arg s] and sink [arg t], finds a [sectref {Flow Problems} "blocking
flow"], which can be used to obtain a [term "maximum flow"] for that [term network] [arg G].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Directed graph [arg G] representing the flow network. Each edge should have attribute
[term throughput] set with integer value.
[arg_def {Node} s input]
The source node for the flow network [arg G].
[arg_def {Node} t input]
The sink node for the flow network [arg G].

[list_end][comment {-- arguments --}]

[def Result:]
Algorithm returns dictionary containing it's blocking flow value for each edge (key) in network [arg G].

[list_end][comment {-- definitions --}]

[emph Note:] Algorithm's complexity is [term O(n^2)], where [term n] is the number of nodes in flow network [arg G].


[call [cmd struct::graph::op::createResidualGraph] [arg G] [arg f]]


Procedure creates a [term "residual graph"] (or [sectref {Flow Problems} "residual network"] ) for network [arg G] and given
flow [arg f].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Flow network (directed graph where each edge has set attribute: [term throughput] ).

[arg_def {dictionary} f input]
Current flows in flow network [arg G].

[list_end][comment {-- arguments --}]

[def Result:]
Procedure returns graph structure that is a [term "residual graph"] created from input flow
network [arg G].
[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::createAugmentingNetwork] [arg G] [arg f] [arg path]]

Procedure creates an [sectref {Flow Problems} "augmenting network"] for a given residual network [arg G]
, flow [arg f] and augmenting path [arg path].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Residual network (directed graph), where for every edge there are set two attributes: throughput and cost.
[arg_def {Dictionary} f input]
Dictionary which contains for every edge (key), current value of the flow on that edge.
[arg_def {List} path input]
Augmenting path, set of edges (list) for which we create the network modification.

[list_end][comment {-- arguments --}]

[def Result:]
Algorithm returns graph structure containing the modified augmenting network.

[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::createLevelGraph] [arg Gf] [arg s]]

For given residual graph [arg Gf] procedure finds the [sectref {Flow Problems} "level graph"].

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} Gf input]
Residual network, where each edge has it's attribute [term throughput] set with certain value.
[arg_def {Node} s input]
The source node for the residual network [arg Gf].
[list_end][comment {-- arguments --}]

[def Result:]
Procedure returns a [term "level graph"] created from input [term "residual network"].
[list_end][comment {-- definitions --}]


[call [cmd struct::graph::op::TSPLocalSearching] [arg G] [arg C]]

Algorithm is a [term "heuristic of local searching"] for [term "Travelling Salesman Problem"]. For some
solution of [term "TSP problem"], it checks if it's possible to find a better solution. As [term "TSP"]
is well known NP-Complete problem, so algorithm is a approximation algorithm (with 2 approximation factor).

[para]
[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Undirected and complete graph with attributes "weight" set on each single edge.
[arg_def {List} C input]
A list of edges being [term "Hamiltonian cycle"], which is solution of [term "TSP Problem"] for graph [arg G].
[list_end][comment {-- arguments --}]

[def Result:]
Algorithm returns the best solution for [term "TSP problem"], it was able to find.
[list_end][comment {-- definitions --}]

[emph Note:] The solution depends on the choosing of the beginning cycle [arg C]. It's not true that better cycle
assures that better solution will be found, but practise shows that we should give starting cycle with as small
sum of weights as possible.


[call [cmd struct::graph::op::TSPLocalSearching3Approx] [arg G] [arg C]]

Algorithm is a [term "heuristic of local searching"] for [term "Travelling Salesman Problem"]. For some
solution of [term "TSP problem"], it checks if it's possible to find a better solution. As [term "TSP"]
is well known NP-Complete problem, so algorithm is a approximation algorithm (with 3 approximation factor).

[para]

[list_begin definitions]

[def Arguments:]
[list_begin arguments]
[arg_def {Graph Object} G input]
Undirected and complete graph with attributes "weight" set on each single edge.
[arg_def {List} C input]
A list of edges being [term "Hamiltonian cycle"], which is solution of [term "TSP Problem"] for graph [arg G].
[list_end][comment {-- arguments --}]

[def Result:]
Algorithm returns the best solution for [term "TSP problem"], it was able to find.

[list_end][comment {-- definitions --}]

[emph Note:] In practise 3-approximation algorithm turns out to be far more effective than 2-approximation, but it gives
worser approximation factor. Further heuristics of local searching (e.g. 4-approximation) doesn't give enough boost to
square the increase of approximation factor, so 2 and 3 approximations are mainly used.


[call [cmd struct::graph::op::createSquaredGraph] [arg G]]

X-Squared graph is a graph with the same set of nodes as input graph [arg G], but a different set of edges. X-Squared graph
has edge [term (u,v)], if and only if, the distance between [term u] and [term v] nodes is not greater than X and [term "u != v"].
[para]
Procedure for input graph [arg G], returns its two-squared graph.

[para]

[emph Note:] Distances used in choosing new set of edges are considering the number of edges, not the sum of weights at edges.


[call [cmd struct::graph::op::createCompleteGraph] [arg G] [arg originalEdges]]

For input graph [arg G] procedure adds missing arcs to make it a [term "complete graph"]. It also holds in
variable [arg originalEdges] the set of arcs that graph [arg G] possessed before that operation.

[list_end]


[section "Background theory and terms"]


[subsection "Shortest Path Problem"]

[list_begin definitions]

[def "Definition ([term "single-pair shortest path problem"]):"]
Formally, given a weighted graph (let [term V] be the set of vertices, and [term E] a set of edges),
and one vertice [term v] of [term V], find a path [term P] from [term v] to a [term "v'"] of V so that
the sum of weights on edges along the path is minimal among all paths connecting v to v'.

[def "Generalizations:"]

[list_begin itemized]
[item][term "The single-source shortest path problem"], in which we have to find shortest paths from a source vertex v to all other vertices in the graph.
[item][term "The single-destination shortest path problem"], in which we have to find shortest paths from all vertices in the graph to a single destination vertex v. This can be reduced to the single-source shortest path problem by reversing the edges in the graph.
[item][term "The all-pairs shortest path problem"], in which we have to find shortest paths between every pair of vertices v, v' in the graph.
[list_end][comment {-- itemized --}]

[emph "Note:"]
The result of [term "Shortest Path problem"] can be [term "Shortest Path tree"], which is a subgraph of a given (possibly weighted) graph constructed so that the
distance between a selected root node and all other nodes is minimal. It is a tree because if there are two paths between the root node and some
vertex v (i.e. a cycle), we can delete the last edge of the longer path without increasing the distance from the root node to any node in the subgraph.

[list_end][comment {-- definitions --}]


[subsection "Travelling Salesman Problem"]

[list_begin definitions]

[def "Definition:"]
For given edge-weighted (weights on edges should be positive) graph the goal is to find the cycle that visits each node in graph
exactly once ([term "Hamiltonian cycle"]).

[def "Generalizations:"]

[list_begin itemized]
[item][term "Metric TSP"] - A very natural restriction of the [term TSP] is to require that the distances between cities form a [term metric], i.e.,
they satisfy [term "the triangle inequality"]. That is, for any 3 cities [term A], [term B] and [term C], the distance between [term A] and [term C]
must be at most the distance from [term A] to [term B] plus the distance from [term B] to [term C]. Most natural instances of [term TSP]
satisfy this constraint.

[item][term "Euclidean TSP"] - Euclidean TSP, or [term "planar TSP"], is the [term TSP] with the distance being the ordinary [term "Euclidean distance"].
[term "Euclidean TSP"] is a particular case of [term TSP] with [term "triangle inequality"], since distances in plane obey triangle inequality. However,
it seems to be easier than general [term TSP] with [term "triangle inequality"]. For example, [term "the minimum spanning tree"] of the graph associated
with an instance of [term "Euclidean TSP"] is a [term "Euclidean minimum spanning tree"], and so can be computed in expected [term "O(n log n)"] time for
[term n] points (considerably less than the number of edges). This enables the simple [term "2-approximation algorithm"] for TSP with triangle
inequality above to operate more quickly.

[item][term "Asymmetric TSP"] - In most cases, the distance between two nodes in the [term TSP] network is the same in both directions.
The case where the distance from [term A] to [term B] is not equal to the distance from [term B] to [term A] is called [term "asymmetric TSP"].
A practical application of an [term "asymmetric TSP"] is route optimisation using street-level routing (asymmetric due to one-way streets,
slip-roads and motorways).

[list_end][comment {-- itemized --}]
[list_end][comment {-- definitions --}]

[subsection "Matching Problem"]

[list_begin definitions]

[def "Definition:"]
Given a graph [term "G = (V,E)"], a matching or [term "edge-independent set"] [term M] in [term G] is a set of pairwise non-adjacent edges,
that is, no two edges share a common vertex. A vertex is [term matched] if it is incident to an edge in the [term "matching M"].
Otherwise the vertex is [term unmatched].


[def "Generalizations:"]

[list_begin itemized]
[item][term "Maximal matching"] - a matching [term M] of a graph G with the property that if any edge not in [term M] is added to [term M],
it is no longer a [term matching], that is, [term M] is maximal if it is not a proper subset of any other [term matching] in graph G.
In other words, a [term "matching M"] of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in [term M].

[item][term "Maximum matching"] - a matching that contains the largest possible number of edges. There may be many [term "maximum matchings"].
The [term "matching number"] of a graph G is the size of a [term "maximum matching"]. Note that every [term "maximum matching"] is [term maximal],
but not every [term "maximal matching"] is a [term "maximum matching"].

[item][term "Perfect matching"] - a matching which matches all vertices of the graph. That is, every vertex of the graph is incident to exactly one
edge of the matching. Every [term "perfect matching"] is [term maximum] and hence [term maximal]. In some literature, the term [term "complete matching"]
is used. A [term "perfect matching"] is also a [term "minimum-size edge cover"]. Moreover, the size of a [term "maximum matching"] is no larger than the
size of a [term "minimum edge cover"].

[item][term "Near-perfect matching"] - a matching in which exactly one vertex is unmatched. This can only occur when the graph has an odd number of vertices,
and such a [term matching] must be [term maximum]. If, for every vertex in a graph, there is a near-perfect matching that omits only that vertex, the graph
is also called [term factor-critical].

[list_end][comment {-- itemized --}]

[def "Related terms:"]

[list_begin itemized]
[item][term "Alternating path"] - given a matching [term M], an [term "alternating path"] is a path in which the edges belong alternatively
to the matching and not to the matching.

[item][term "Augmenting path"] - given a matching [term M], an [term "augmenting path"] is an [term "alternating path"] that starts from
and ends on free (unmatched) vertices.

[list_end][comment {-- itemized --}]

[list_end][comment {-- definitons --}]


[subsection "Cut Problems"]

[list_begin definitions]

[def "Definition:"]
A [term cut] is a partition of the vertices of a graph into two [term "disjoint subsets"]. The [term cut-set] of the [term cut] is the
set of edges whose end points are in different subsets of the partition. Edges are said to be crossing the cut if they are in its [term cut-set].
[para]
Formally:
[list_begin itemized]
[item] a [term cut] [term "C = (S,T)"] is a partition of [term V] of a graph [term "G = (V, E)"].

[item] an [term "s-t cut"] [term "C = (S,T)"] of a [term "flow network"] [term "N = (V, E)"] is a cut of [term N] such that [term s] is included in [term S]
and [term t] is included in [term T], where [term s] and [term t] are the [term source] and the [term sink] of [term N] respectively.

[item] The [term cut-set] of a [term "cut C = (S,T)"] is such set of edges from graph [term "G = (V, E)"] that each edge [term "(u, v)"] satisfies
condition that [term u] is included in [term S] and [term v] is included in [term T].

[list_end][comment {-- itemized --}]
[para]
In an [term "unweighted undirected"] graph, the size or weight of a cut is the number of edges crossing the cut. In a [term "weighted graph"],
the same term is defined by the sum of the weights of the edges crossing the cut.
[para]
In a [term "flow network"], an [term "s-t cut"] is a cut that requires the [term source] and the [term sink] to be in different subsets,
and its [term cut-set] only consists of edges going from the [term source's] side to the [term sink's] side. The capacity of an [term "s-t cut"]
is defined by the sum of capacity of each edge in the [term cut-set].
[para]
The [term cut] of a graph can sometimes refer to its [term cut-set] instead of the partition.


[def "Generalizations:"]
[list_begin itemized]
[item][term "Minimum cut"] - A cut is minimum if the size of the cut is not larger than the size of any other cut.
[item][term "Maximum cut"] - A cut is maximum if the size of the cut is not smaller than the size of any other cut.
[item][term "Sparsest cut"] - The [term "Sparsest cut problem"] is to bipartition the vertices so as to minimize the ratio of the number
of edges across the cut divided by the number of vertices in the smaller half of the partition.
[list_end][comment {-- itemized --}]

[list_end][comment {-- definitons --}]


[subsection "K-Center Problem"]

[list_begin definitions]

[def "Definitions:"]
[list_begin definitions]
[def "[term "Unweighted K-Center"]"]
For any set [term S] ( which is subset of [term V] ) and node [term v], let the [term connect(v,S)] be the
cost of cheapest edge connecting [term v] with any node in [term S]. The goal is to find
such [term S], that [term "|S| = k"] and [term "max_v{connect(v,S)}"] is possibly small.
[para]
In other words, we can use it i.e. for finding best locations in the city ( nodes
of input graph ) for placing k buildings, such that those buildings will be as close
as possible to all other locations in town.
[para]
[def "[term "Weighted K-Center"]"]
The variation of [term "unweighted k-center problem"]. Besides the fact graph is edge-weighted,
there are also weights on vertices of input graph [arg G]. We've got also restriction
[arg W]. The goal is to choose such set of nodes [term S] ( which is a subset of [term V] ), that it's
total weight is not greater than [arg W] and also function: [term "max_v { min_u { cost(u,v) }}"]
has the smallest possible worth ( [term v] is a node in [term V] and [term u] is a node in [term S] ).

[list_end][comment {-- definitions --}]
[list_end][comment {-- definitions --}]


[subsection "Flow Problems"]

[list_begin definitions]

[def "Definitions:"]
[list_begin itemized]
[item][term "the maximum flow problem"] - the goal is to find a feasible flow through a single-source, single-sink flow network that is maximum.
The [term "maximum flow problem"] can be seen as a special case of more complex network flow problems, such as the [term "circulation problem"].
The maximum value of an [term "s-t flow"] is equal to the minimum capacity of an [term "s-t cut"] in the network, as stated in the
[term "max-flow min-cut theorem"].
[para]
More formally for flow network [term "G = (V,E)"], where for each edge [term "(u, v)"] we have its throuhgput [term "c(u,v)"] defined. As [term flow]
[term F] we define set of non-negative integer attributes [term f(u,v)] assigned to edges, satisfying such conditions:
[list_begin enumerated]
[enum]for each edge [term "(u, v)"] in [term G] such condition should be satisfied:      0 <= f(u,v) <= c(u,v)
[enum]Network [term G] has source node [term s] such that the flow [term F] is equal to the sum of outcoming flow decreased by the sum of incoming flow from that source node [term s].
[enum]Network [term G] has sink node [term t] such that the the [term -F] value is equal to the sum of the incoming flow decreased by the sum of outcoming flow from that sink node [term t].
[enum]For each node that is not a [term source] or [term sink] the sum of incoming flow and sum of outcoming flow should be equal.
[list_end][comment {-- enumerated --}]

[item][term "the minimum cost flow problem"] - the goal is finding the cheapest possible way of sending a certain amount of flow through a [term "flow network"].

[item][term "blocking flow"] - a [term "blocking flow"] for a [term "residual network"] [term Gf] we name such flow [term b] in [term Gf] that:
[list_begin enumerated]
[enum]Each path from [term sink] to [term source] is the shortest path in [term Gf].
[enum]Each shortest path in [term Gf] contains an edge with fully used throughput in [term Gf+b].
[list_end][comment {-- enumerated --}]

[item][term "residual network"] - for a flow network [term G] and flow [term f] [term "residual network"] is built with those edges, which can
send larger flow. It contains only those edges, which can send flow larger than 0.

[item][term "level network"] - it has the same set of nodes as [term "residual graph"], but has only those edges [term "(u,v)"] from [arg Gf]
for which such equality is satisfied: [term "distance(s,u)+1 = distance(s,v)"].

[item][term "augmenting network"] - it is a modification of [term "residual network"] considering the new
flow values. Structure stays unchanged but values of throughputs and costs at edges
are different.

[list_end][comment {-- itemized --}]
[list_end][comment {-- definitions --}]


[subsection "Approximation algorithm"]

[list_begin definitions]

[def "k-approximation algorithm:"]
Algorithm is a k-approximation, when for [term ALG] (solution returned by algorithm) and
[term OPT] (optimal solution), such inequality is true:

[list_begin itemized]
[item] for minimalization problems: [term "ALG/OPT <= k" ]
[item] for maximalization problems: [term "OPT/ALG <= k" ]
[list_end][comment {-- itemized --}]


[list_end][comment {-- definitions --}]


[section References]

[list_begin enum]
[enum] [uri http://en.wikipedia.org/wiki/Adjacency_matrix      {Adjacency matrix}]
[enum] [uri http://en.wikipedia.org/wiki/Adjacency_list		   {Adjacency list}]
[enum] [uri http://en.wikipedia.org/wiki/Kruskal%27s_algorithm {Kruskal's algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Prim%27s_algorithm    {Prim's algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Bipartite_graph       {Bipartite graph}]
[enum] [uri http://en.wikipedia.org/wiki/Strongly_connected_components {Strongly connected components}]
[enum] [uri http://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm {Tarjan's strongly connected components algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Cut_vertex {Cut vertex}]
[enum] [uri http://en.wikipedia.org/wiki/Bridge_(graph_theory) Bridge]
[enum] [uri http://en.wikipedia.org/wiki/Bellman-Ford_algorithm {Bellman-Ford's algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Johnson_algorithm {Johnson's algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Floyd-Warshall_algorithm {Floyd-Warshall's algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Travelling_salesman_problem {Travelling Salesman Problem}]
[enum] [uri http://en.wikipedia.org/wiki/Christofides_algorithm {Christofides Algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Maxcut {Max Cut}]
[enum] [uri http://en.wikipedia.org/wiki/Matching {Matching}]
[enum] [uri http://en.wikipedia.org/wiki/Maximal_independent_set {Max Independent Set}]
[enum] [uri http://en.wikipedia.org/wiki/Vertex_cover_problem {Vertex Cover}]
[enum] [uri http://en.wikipedia.org/wiki/Ford-Fulkerson_algorithm {Ford-Fulkerson's algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Maximum_flow_problem {Maximum Flow problem}]
[enum] [uri http://en.wikipedia.org/wiki/Minimum_cost_flow_problem {Busacker-Gowen's algorithm}]
[enum] [uri http://en.wikipedia.org/wiki/Dinic's_algorithm {Dinic's algorithm}]
[enum] [uri http://www.csc.kth.se/~viggo/wwwcompendium/node128.html {K-Center problem}]
[enum] [uri http://en.wikipedia.org/wiki/Breadth-first_search {BFS}]
[enum] [uri http://en.wikipedia.org/wiki/Degree-constrained_spanning_tree {Minimum Degree Spanning Tree}]
[enum] [uri http://en.wikipedia.org/wiki/Approximation_algorithm {Approximation algorithm}]
[list_end]


[section {BUGS, IDEAS, FEEDBACK}]

This document, and the package it describes, will undoubtedly contain
bugs and other problems.

Please report such in the category [emph {struct :: graph}] of the
[uri {http://sourceforge.net/tracker/?group_id=12883} {Tcllib SF Trackers}].

Please also report any ideas for enhancements you may have for either
package and/or documentation.


[keywords graph dijkstra distance radius diameter eccentricity]
[keywords edge arc node vertex subgraph neighbour]
[keywords adjacent loop degree]
[keywords {cut vertex} {articulation point} {connected component}]
[keywords {strongly connected component} {adjacency matrix} {adjacency list}]
[keywords {minimal spanning tree} bipartite bridge {cut edge} isthmus]
[keywords {shortest path} {travelling salesman} {max cut} matching {independent set}]
[keywords {vertex cover} {maximum flow} {minimum cost flow} {blocking flow}]
[keywords {augmenting path} {residual graph} {level graph} {flow network} {augmenting network}]
[keywords {complete graph} {squared graph} {local searching} {heuristic} {bfs} {approximation algorithm}]
[keywords {minimum diameter spanning tree} {minimum degree spanning tree} {degree constrained spanning tree}]
[manpage_end]